FPS の一括入出力

components/fps_io.hpp · GitHub で見る

共通ページ

法 998244353 の共有 mint の係数列をまとめて読み書きする部品です。出力には FPS の一括出力を読み込みます。出力だけが必要なら components/fps_write.hpp を直接 include すると、入力処理を読み込まずに使えます。

read_coefficients は Linux x86_64 / GCC 用に定義されます。呼び出し側で AVX2 対応を確認してから使ってください。他の環境では FastInput::read と mint::raw などで係数を読みます。write_coefficients は Mac を含むすべての対応環境で使え、CPU の選択は自動です。

操作	内容・返り値	計算量
`fps_detail::read_coefficients(input,out,n)`	`FastInput& input` から `size_t n` 個を読み、`vector<mint>& out` の末尾へ追加する。返り値は `void`	\(O(L+m)\)
`fps_detail::write_coefficients(output,in,n)`	`FastOutput& output` へ、`const mint* in` の先頭 `size_t n` 係数を出力する。返り値は `void`	\(O(n)\)

既定引数はありません。表の \(L\) は読み進める文字数、\(m\) は呼び出し前の out.size() です。容量が足りている場合、読み込みは \(O(L)\) 時間です。通常の整数入出力には Fast I/O の read・write も使えます。

read_coefficients(input,out,n) は、入力・追加先・係数数の順に指定します。空白区切りの十進整数が n 個以上存在し、各値が uint32_t に収まることを前提とします。各整数を法 998244353 の値として読み、既存の out の要素を保持して末尾へ追加します。零や同じ係数があっても入力順を保ちます。n=0 は何も追加せず、入力も消費しません。指定した個数の後のトークンは、続けて input.read(x) などで読めます。

大きな入力ではトークン境界で 4 ストリームへ分け、各ストリームから 2 トークンずつ並行して変換します。すべての桁・区切り・uint32_t の上限を確認してから結果を採用します。入力の末尾や不規則な空白では既存の整数読み込みに戻ります。バッファの外を読まず、利用者の入力に padding は要求しません。変換した整数は 64 個ずつ Montgomery 表現へ変換して追加します。

n > out.max_size()-out.size() なら、読み進める前に std::length_error を送出します。入力元の寿命は読み込みが終わるまで保ち、同じ FastInput を呼び出し中に別の処理から使わないでください。

write_coefficients(output,in,n) の引数順序は出力先・係数列の先頭・係数数です。入力を変更せず、アラインメントや末尾の padding も要求しません。n=0 は in=nullptr でもよく、改行だけを出力します。出力書式・エラー処理・使用例は FPS の一括出力にまとめています。

入力・出力ともに、配列以外の追加作業空間は \(O(1)\) です。入力先の再確保が必要な場合は \(O(m+n)\) の空間を使います。作業領域は呼び出しごとに所有し、共有する可変バッファはありません。

依存ライブラリ

検証コード

コード

#line 1 "components/fps_io.hpp"

#line 1 "components/fast_io_bulk.hpp"

#line 1 "components/fast_io_output.hpp"

#line 1 "components/fast_io.hpp"

#include <algorithm>
#include <array>
#include <bit>
#include <cassert>
#include <charconv>
#include <cstddef>
#include <cstdint>
#include <cstdio>
#include <cstring>
#include <memory>
#include <stdexcept>
#include <string>
#include <string_view>
#include <type_traits>
#if defined(__unix__) || defined(__APPLE__)
#include <sys/mman.h>
#include <sys/stat.h>
#include <unistd.h>
#endif

namespace fast_io_detail { struct BulkInput; struct BulkOutput; }

class FastInput {
	friend struct fast_io_detail::BulkInput;
	static constexpr std::size_t capacity = 1 << 16;
	FILE *file = nullptr;
	std::unique_ptr<char[]> buffer;
	const char *current = "", *end = current;
	std::string scratch;
	void *mapping = nullptr;
	std::size_t mapped_size = 0;
	bool refill() {
		if (!file) return false;
		std::size_t n = std::fread(buffer.get(), 1, capacity, file);
		current = buffer.get();
		end = current + n;
		return n;
	}
	static const char *scan(const char *p, const char *end) {
		while (end - p >= 8) {
			std::uint64_t x;
			std::memcpy(&x, p, 8);
			if ((x - 0x2121212121212121ULL) & ~x & 0x8080808080808080ULL) break;
			p += 8;
		}
		while (p != end && static_cast<unsigned char>(*p) > ' ') ++p;
		return p;
	}

public:
	explicit FastInput(FILE *input = stdin) {
#if defined(__unix__) || defined(__APPLE__)
		struct stat status;
		int fd = fileno(input);
		auto offset = ftello(input);
		if (fd >= 0 && offset >= 0 && !fstat(fd, &status) && S_ISREG(status.st_mode) && status.st_size > offset) {
			mapped_size = status.st_size;
			mapping = mmap(nullptr, mapped_size, PROT_READ, MAP_PRIVATE, fd, 0);
			if (mapping != MAP_FAILED) {
				current = static_cast<const char *>(mapping) + offset;
				end = static_cast<const char *>(mapping) + mapped_size;
				return;
			}
			mapping = nullptr;
		}
#endif
		file = input;
		buffer = std::make_unique_for_overwrite<char[]>(capacity);
	}
	explicit FastInput(std::string_view input) : current(input.empty() ? "" : input.data()), end(current + input.size()) {}
	FastInput(const FastInput &) = delete;
	FastInput &operator=(const FastInput &) = delete;
	~FastInput() {
#if defined(__unix__) || defined(__APPLE__)
		if (mapping) munmap(mapping, mapped_size);
#endif
	}
	std::string_view token() {
		while (true) {
			while (current != end && static_cast<unsigned char>(*current) <= ' ') ++current;
			if (current != end) break;
			if (!refill()) return {};
		}
		const char *begin = current;
		current = scan(current, end);
		if (current != end || !file) return {begin, std::size_t(current - begin)};
		scratch.assign(begin, end);
		while (refill()) {
			current = scan(current, end);
			scratch.append(buffer.get(), std::size_t(current - buffer.get()));
			if (current != end) break;
		}
		return scratch;
	}
	template <class T>
	bool read(T &value) {
		if constexpr (std::is_integral_v<T> && !std::is_same_v<T, bool>) {
			while (true) {
				while (current != end && static_cast<unsigned char>(*current) <= ' ') ++current;
				if (current != end) break;
				if (!refill()) return false;
			}
			if constexpr (std::is_unsigned_v<T> && sizeof(T) == 4 && std::endian::native == std::endian::little) {
				if (end - current >= 11) {
					std::uint64_t x;
					std::memcpy(&x, current, 8);
					if (!(((x + 0x4646464646464646ULL) | (x - 0x3030303030303030ULL)) & 0x8080808080808080ULL)) {
						x &= 0x0f0f0f0f0f0f0f0fULL;
						x = (x * 10 + (x >> 8)) & 0x00ff00ff00ff00ffULL;
						x = (x * 100 + (x >> 16)) & 0x0000ffff0000ffffULL;
						std::uint64_t v = std::uint32_t(x * 10000 + (x >> 32));
						int n = 8;
						for (; n < 10 && static_cast<unsigned char>(current[n]) - unsigned('0') < 10; ++n)
							v = v * 10 + (current[n] - '0');
						if (static_cast<unsigned char>(current[n]) <= ' ' && v <= 0xffffffffULL) {
							current += n;
							value = T(v);
							return true;
						}
					}
				}
			}
			T parsed{};
			auto result = std::from_chars(current, end, parsed);
			if (result.ec == std::errc{} && (result.ptr != end ? static_cast<unsigned char>(*result.ptr) <= ' ' : !file)) {
				current = result.ptr;
				value = parsed;
				return true;
			}
		}
		auto s = token();
		if (s.empty()) return false;
		if constexpr (std::is_same_v<T, bool>) {
			assert(s == "0" || s == "1");
			value = s[0] == '1';
		} else if constexpr (std::is_integral_v<T>) {
			auto result = std::from_chars(s.data(), s.data() + s.size(), value);
			assert(result.ec == std::errc{} && result.ptr == s.data() + s.size());
			(void)result;
		} else value = T(s);
		return true;
	}
};

class FastOutput {
	friend struct fast_io_detail::BulkOutput;
	inline static constexpr auto quads = [] {
		std::array<std::array<char, 4>, 10000> table{};
		for (int i = 0; i < 10000; ++i)
			table[i] = {char('0' + i / 1000), char('0' + i / 100 % 10), char('0' + i / 10 % 10), char('0' + i % 10)};
		return table;
	}();
	FILE *file;
	std::unique_ptr<char[]> buffer = std::make_unique_for_overwrite<char[]>(1 << 16);
	std::size_t capacity = 1 << 16, length = 0;
	char *reserve(std::size_t n) {
		if (n > capacity - length) {
			if (!flush()) throw std::runtime_error("output failed");
			if (n > capacity) {
				std::size_t next = std::max(n, 2 * capacity);
				buffer = std::make_unique_for_overwrite<char[]>(next);
				capacity = next;
			}
		}
		return buffer.get() + length;
	}

public:
	explicit FastOutput(FILE *output = stdout) : file(output) {}
	FastOutput(const FastOutput &) = delete;
	FastOutput &operator=(const FastOutput &) = delete;
	~FastOutput() { flush(); }
	bool flush() {
		std::size_t offset = 0;
		while (offset < length) {
			auto n = std::fwrite(buffer.get() + offset, 1, length - offset, file);
			if (!n) {
				std::memmove(buffer.get(), buffer.get() + offset, length - offset);
				length -= offset;
				return false;
			}
			offset += n;
		}
		length = 0;
		return std::fflush(file) == 0;
	}
	void write(char c) { *reserve(1) = c; ++length; }
	void write(__uint128_t value) {
		char digits[40];
		char *end = digits + sizeof(digits), *begin = end;
		do { *--begin = char('0' + value % 10); value /= 10; } while (value);
		write(std::string_view(begin, end - begin));
	}
	void write(__int128_t value) {
		__uint128_t magnitude = __uint128_t(value);
		if (value < 0) write('-'), magnitude = 0 - magnitude;
		write(magnitude);
	}
	void write(std::string_view s) {
		char *out = reserve(s.size());
		if (!s.empty()) std::memcpy(out, s.data(), s.size());
		length += s.size();
	}
	template <class T> requires std::is_integral_v<T>
	void write(T value) {
		char *out = reserve(32);
		if constexpr (std::is_same_v<T, bool>) *out = '0' + value, ++length;
		else if constexpr (std::is_unsigned_v<T> && sizeof(T) == 4) {
			char *p = out;
			if (value >= 100000000) {
				auto high = value / 100000000, low = value % 100000000;
				p = std::to_chars(p, p + 2, high).ptr;
				std::memcpy(p, quads[low / 10000].data(), 4);
				std::memcpy(p + 4, quads[low % 10000].data(), 4);
				p += 8;
			} else if (value >= 10000) {
				p = std::to_chars(p, p + 4, value / 10000).ptr;
				std::memcpy(p, quads[value % 10000].data(), 4);
				p += 4;
			} else p = std::to_chars(p, p + 4, value).ptr;
			length += p - out;
		} else length += std::to_chars(out, out + 32, value).ptr - out;
	}
	template <class T> requires requires(const T &value, char *out) { value.decimal_capacity(); value.to_chars(out); }
	void write(const T &value) {
		char *out = reserve(value.decimal_capacity());
		length += value.to_chars(out) - out;
	}
	template <class T>
	void println(const T &value) { write(value); write('\n'); }
};

#line 5 "components/fast_io_output.hpp"
#include <limits>
#include <utility>

namespace fast_io_detail {
struct BulkOutput {
	static char *small(char *out, std::uint32_t value) {
		unsigned n = 1 + (value >= 10) + (value >= 100) + (value >= 1000);
		if constexpr (std::endian::native == std::endian::little) {
			std::uint32_t word;
			std::memcpy(&word, FastOutput::quads[value].data(), 4);
			word >>= (4 - n) * 8;
			std::memcpy(out, &word, 4);
		} else std::memcpy(out, FastOutput::quads[value].data() + 4 - n, n);
		return out + n;
	}
	static char *number(char *out, std::uint32_t value) {
		if (value >= 100000000) {
			out = small(out, value / 100000000);
			value %= 100000000;
			std::memcpy(out, FastOutput::quads[value / 10000].data(), 4);
			std::memcpy(out + 4, FastOutput::quads[value % 10000].data(), 4);
			return out + 8;
		}
		if (value >= 10000) {
			out = small(out, value / 10000);
			std::memcpy(out, FastOutput::quads[value % 10000].data(), 4);
			return out + 4;
		}
		return small(out, value);
	}
	template<std::size_t Fields, class Read>
#if defined(__GNUC__) || defined(__clang__)
	[[gnu::noinline]]
#endif
	static void records(FastOutput &out, std::size_t count, Read read) {
		constexpr auto bytes = 11 * Fields;
		constexpr auto block = std::max<std::size_t>(1, (1 << 16) / bytes);
		for (std::size_t base = 0; base < count;) {
			const auto n = std::min(count - base, block);
			char *begin = out.reserve(n * bytes), *p = begin;
			for (std::size_t i = 0; i < n; ++i) {
				auto values = read(base + i);
				for (std::size_t j = 0; j < Fields; ++j) {
					p = number(p, std::uint32_t(values[j]));
					*p++ = j + 1 == Fields ? '\n' : ' ';
				}
			}
			out.length += p - begin;
			base += n;
		}
	}
	static void write(FastOutput &out, std::string_view text) {
		if (text.size() <= (1 << 16)) return out.write(text);
		if (out.length && !out.flush()) throw std::runtime_error("output failed");
		std::size_t done = 0;
		while (done < text.size()) {
			const auto written = std::fwrite(text.data() + done, 1, text.size() - done, out.file);
			if (!written) throw std::runtime_error("output failed");
			done += written;
		}
	}
};

}

template<std::size_t Fields, class Read>
void write_unsigned_records(FastOutput &output, std::size_t count, Read read) {
	static_assert(Fields > 0 && Fields <= std::numeric_limits<std::size_t>::max() / 11);
	fast_io_detail::BulkOutput::records<Fields>(output, count, std::move(read));
}

#line 6 "components/fast_io_bulk.hpp"
#include 
#line 8 "components/fast_io_bulk.hpp"

#if defined(__linux__) && defined(__x86_64__) && defined(__GNUC__) && !defined(__clang__)
#include <immintrin.h>
#pragma GCC push_options
_Pragma("GCC target(\"avx2\")")
#pragma GCC optimize("O3")
namespace fast_io_detail {
struct BulkInput {
	alignas(32) static inline constexpr auto rows = [] {
		std::array<std::array<unsigned char, 16>, 11> r{};
		for (int n = 0; n <= 10; ++n)
			for (int i = 0; i < 16; ++i) r[n][i] = i < 16 - n ? 128 : i - (16 - n);
		return r;
	}();
	static constexpr std::size_t kChunk = 1 << 17, kCapacity = kChunk / 4 + 66;

static unsigned separators(__m256i bytes) {
		return unsigned(_mm256_movemask_epi8(_mm256_cmpeq_epi8(_mm256_max_epu8(bytes, _mm256_set1_epi8(' ')), _mm256_set1_epi8(' '))));
	}

[[gnu::always_inline]] static __m256i two(const char *&p, __m256i &invalid) {
		const auto bytes = _mm256_loadu_si256(reinterpret_cast<const __m256i *>(p));
		const unsigned sep = separators(bytes), a = std::countr_zero(sep), b = std::countr_zero(sep & (sep - 1));
		if (a < 1 || a > 10 || b <= a + 1 || b - a - 1 > 10) [[unlikely]] {
			invalid = _mm256_set1_epi32(-1);
			p += std::min(b + 1, 32u);
			return _mm256_setzero_si256();
		}
		const auto window = _mm256_inserti128_si256(bytes, _mm_loadu_si128(reinterpret_cast<const __m128i *>(p + a + 1)), 1);
		const auto control = _mm256_set_m128i(_mm_loadu_si128(reinterpret_cast<const __m128i *>(rows[b - a - 1].data())),
			_mm_loadu_si128(reinterpret_cast<const __m128i *>(rows[a].data())));
		const auto digits = _mm256_shuffle_epi8(_mm256_sub_epi8(window, _mm256_set1_epi8('0')), control);
		invalid = _mm256_or_si256(invalid, _mm256_subs_epu8(digits, _mm256_set1_epi8(9)));
		p += b + 1;
		return _mm256_madd_epi16(_mm256_maddubs_epi16(digits, _mm256_set1_epi16(0x010a)), _mm256_set1_epi32(0x00010064));
	}

static std::uint32_t one(const char *&p, bool &bad) {
		const auto bytes = _mm_loadu_si128(reinterpret_cast<const __m128i *>(p));
		const auto spaces = _mm_cmpeq_epi8(_mm_max_epu8(bytes, _mm_set1_epi8(' ')), _mm_set1_epi8(' '));
		const unsigned n = std::countr_zero(unsigned(_mm_movemask_epi8(spaces)) | 0x10000u);
		if (n < 1 || n > 10) [[unlikely]] {
			bad = true;
			p += std::min(n + 1, 16u);
			return 0;
		}
		const auto digits = _mm_shuffle_epi8(_mm_sub_epi8(bytes, _mm_set1_epi8('0')),
			_mm_loadu_si128(reinterpret_cast<const __m128i *>(rows[n].data())));
		const auto invalid = _mm_subs_epu8(digits, _mm_set1_epi8(9));
		bad |= !_mm_testz_si128(invalid, invalid);
		const auto groups = _mm_madd_epi16(_mm_maddubs_epi16(digits, _mm_set1_epi16(0x010a)), _mm_set1_epi32(0x00010064));
		const auto halves = _mm_madd_epi16(_mm_packus_epi32(groups, groups), _mm_set1_epi32(0x00012710));
		const std::uint64_t value = std::uint64_t(unsigned(_mm_cvtsi128_si32(halves))) * 100000000 + unsigned(_mm_extract_epi32(halves, 1));
		bad |= value > 0xffffffffULL;
		p += n + 1;
		return std::uint32_t(value);
	}

static const char *after_separator(const char *p) {
		const unsigned mask = separators(_mm256_loadu_si256(reinterpret_cast<const __m256i *>(p)));
		return mask ? p + std::countr_zero(mask) + 1 : nullptr;
	}

static std::size_t lockstep(const char *(&streams)[4], const char *const (&end)[4], unsigned *const (&out)[4], __m256i &invalid) {
		const char *p[4] = {streams[0], streams[1], streams[2], streams[3]};
		std::size_t done = 0;
		for (;;) {
			std::ptrdiff_t left = end[0] - p[0];
			for (int k = 1; k < 4; ++k) left = std::min(left, end[k] - p[k]);
			const auto steps = std::min(left / 33, std::ptrdiff_t((kCapacity - done) / 2));
			if (steps <= 0) break;
			for (auto j = steps; j; --j, done += 2) {
				const auto g0 = two(p[0], invalid), g1 = two(p[1], invalid), g2 = two(p[2], invalid), g3 = two(p[3], invalid);
				const auto k = _mm256_set1_epi32(0x00012710);
				const auto h01 = _mm256_castsi256_ps(_mm256_madd_epi16(_mm256_packus_epi32(g0, g1), k));
				const auto h23 = _mm256_castsi256_ps(_mm256_madd_epi16(_mm256_packus_epi32(g2, g3), k));
				const auto high = _mm256_castps_si256(_mm256_shuffle_ps(h01, h23, 0x88));
				const auto low = _mm256_castps_si256(_mm256_shuffle_ps(h01, h23, 0xdd));
				const auto overflow = _mm256_or_si256(_mm256_cmpgt_epi32(high, _mm256_set1_epi32(42)),
					_mm256_and_si256(_mm256_cmpeq_epi32(high, _mm256_set1_epi32(42)), _mm256_cmpgt_epi32(low, _mm256_set1_epi32(94967295))));
				invalid = _mm256_or_si256(invalid, overflow);
				const auto value = _mm256_add_epi32(_mm256_mullo_epi32(high, _mm256_set1_epi32(100000000)), low);
				const auto q = _mm256_permutevar8x32_epi32(value, _mm256_setr_epi32(0, 4, 1, 5, 2, 6, 3, 7));
				const auto q01 = _mm256_castsi256_si128(q), q23 = _mm256_extracti128_si256(q, 1);
				_mm_storel_epi64(reinterpret_cast<__m128i *>(out[0] + done), q01);
				_mm_storeh_pd(reinterpret_cast<double *>(out[1] + done), _mm_castsi128_pd(q01));
				_mm_storel_epi64(reinterpret_cast<__m128i *>(out[2] + done), q23);
				_mm_storeh_pd(reinterpret_cast<double *>(out[3] + done), _mm_castsi128_pd(q23));
			}
		}
		for (int k = 0; k < 4; ++k) streams[k] = p[k];
		return done;
	}

template <class Emit>
	static std::size_t chunk_blocks(const char *&p, const char *limit, std::size_t count, unsigned *const (&out)[4], Emit &&emit) {
		const auto bytes = std::min({kChunk, 2 * (count - 32), std::size_t(limit - p) - 64}) & ~std::size_t(3);
		const std::size_t stream = bytes / 4;
		const char *p1 = after_separator(p + stream), *p2 = after_separator(p + 2 * stream);
		const char *p3 = after_separator(p + 3 * stream), *last = after_separator(p + bytes);
		if (!p1 || !p2 || !p3 || !last) return 0;
		const char *s[4] = {p, p1, p2, p3}, *const end[4] = {p1, p2, p3, last};
		auto invalid = _mm256_setzero_si256();
		const auto done = lockstep(s, end, out, invalid);
		bool bad = !_mm256_testz_si256(invalid, invalid);
		std::size_t n[4];
		for (int k = 0; k < 4; ++k) {
			n[k] = done;
			while (s[k] < end[k] && n[k] < kCapacity) out[k][n[k]++] = one(s[k], bad);
			bad |= s[k] != end[k];
		}
		const auto parsed = n[0] + n[1] + n[2] + n[3];
		if (bad || parsed > count) return 0;
		for (int k = 0; k < 4; ++k) emit(out[k], n[k]);
		p = last;
		return parsed;
	}

static std::size_t chunk(const char *&p, const char *limit, unsigned *dst, std::size_t count, unsigned *scratch) {
		unsigned *const out[4] = {dst, scratch, scratch + kCapacity, scratch + 2 * kCapacity};
		bool first = true;
		unsigned *next = dst;
		return chunk_blocks(p, limit, count, out, [&](const unsigned *values, std::size_t n) {
			if (!first) std::memcpy(next, values, n * sizeof(unsigned));
			first = false;
			next += n;
		});
	}

template <class Emit>
	static void read_blocks(FastInput &in, std::size_t count, Emit emit) {
		std::unique_ptr<unsigned[]> scratch;
		std::size_t done = 0;
		bool chunks = true;
		while (done < count) {
			const char *p = in.current;
			while (p != in.end && static_cast<unsigned char>(*p) <= ' ') ++p;
			while (chunks && count - done >= 1024 && in.end - p >= 1088) {
				if (!scratch) scratch = std::make_unique_for_overwrite<unsigned[]>(4 * kCapacity);
				unsigned *const out[4] = {scratch.get(), scratch.get() + kCapacity, scratch.get() + 2 * kCapacity, scratch.get() + 3 * kCapacity};
				const auto parsed = chunk_blocks(p, in.end, count - done, out, emit);
				if (!parsed) {
					chunks = false;
					break;
				}
				done += parsed;
			}
			unsigned tail[64];
			std::size_t used = 0;
			while (done + 2 <= count && in.end - p >= 32) {
				const char *next = p;
				auto invalid = _mm256_setzero_si256();
				const auto groups = two(next, invalid);
				if (!_mm256_testz_si256(invalid, invalid)) break;
				const auto halves = _mm256_madd_epi16(_mm256_packus_epi32(groups, groups), _mm256_set1_epi32(0x00012710));
				const unsigned h0 = _mm256_extract_epi32(halves, 0), l0 = _mm256_extract_epi32(halves, 1);
				const unsigned h1 = _mm256_extract_epi32(halves, 4), l1 = _mm256_extract_epi32(halves, 5);
				if (h0 > 42 || (h0 == 42 && l0 > 94967295) || h1 > 42 || (h1 == 42 && l1 > 94967295)) break;
				tail[used++] = h0 * 100000000 + l0;
				tail[used++] = h1 * 100000000 + l1;
				if (used == 64) {
					emit(tail, used);
					used = 0;
				}
				done += 2;
				p = next;
			}
			if (used) emit(tail, used);
			in.current = p;
			if (done < count) {
				unsigned value = 0;
				in.read(value);
				emit(&value, 1);
				++done;
			}
		}
	}

static void read(FastInput &in, unsigned *out, std::size_t count) {
		read_blocks(in, count, [&](const unsigned *values, std::size_t n) {
			std::memcpy(out, values, n * sizeof(unsigned));
			out += n;
		});
	}

};
}
#pragma GCC pop_options
#endif

namespace fast_io_detail {
template <class Emit, std::size_t... I>
void emit_record(Emit &emit, const std::uint32_t *values, std::index_sequence<I...>) {
	emit(values[I]...);
}
}

template <class Emit>
void read_unsigned_blocks(FastInput &input, std::size_t count, Emit emit) {
#if defined(__linux__) && defined(__x86_64__) && defined(__GNUC__) && !defined(__clang__)
	if (__builtin_cpu_supports("avx2")) {
		fast_io_detail::BulkInput::read_blocks(input, count, emit);
		return;
	}
#endif
	std::uint32_t values[64];
	while (count) {
		const auto n = std::min<std::size_t>(count, 64);
		for (std::size_t i = 0; i < n; ++i) input.read(values[i]);
		emit(values, n);
		count -= n;
	}
}

inline void read_unsigned(FastInput &input, std::span<std::uint32_t> out) {
	read_unsigned_blocks(input, out.size(), [&](const std::uint32_t *values, std::size_t count) {
		std::memcpy(out.data(), values, count * sizeof(std::uint32_t));
		out = out.subspan(count);
	});
}

template <std::size_t Fields, class Emit>
void read_unsigned_records(FastInput &input, std::size_t count, Emit emit) {
	static_assert(Fields > 0);
	if (count > std::numeric_limits<std::size_t>::max() / Fields) throw std::length_error("input size");
	std::array<std::uint32_t, Fields> pending;
	std::size_t used = 0;
	constexpr auto indices = std::make_index_sequence<Fields>();
	read_unsigned_blocks(input, count * Fields, [&](const std::uint32_t *values, std::size_t n) {
		std::size_t i = 0;
		if (used) {
			while (used < Fields && i < n) pending[used++] = values[i++];
			if (used == Fields) {
				fast_io_detail::emit_record(emit, pending.data(), indices);
				used = 0;
			}
		}
		for (; i < n && n - i >= Fields; i += Fields) fast_io_detail::emit_record(emit, values + i, indices);
		while (i < n) pending[used++] = values[i++];
	});
}

#line 1 "components/fps_output.hpp"

#line 1 "components/mint_arithmetic.hpp"

#line 1 "math/mint.hpp"

#line 4 "math/mint.hpp"
#include <istream>
#include <ostream>
#line 7 "math/mint.hpp"

#if defined(__GNUC__) && !defined(__clang__) && (defined(__x86_64__) || defined(__i386__))
#pragma GCC push_options
#pragma GCC optimize("O3")
#endif

namespace mint_detail {
template <class I>
constexpr bool mint_integral = std::is_integral_v
#if defined(__SIZEOF_INT128__)
							 || std::is_same_v<I, __int128> || std::is_same_v<I, unsigned __int128>
#endif
	;
struct [[gnu::warn_unused]] Mint {
	using mint = Mint;
	unsigned v = 0;
	[[nodiscard]] static constexpr unsigned mod() { return 998244353; }
	[[nodiscard, gnu::always_inline]] static unsigned reduce(unsigned long long x) {
		return (x + (unsigned long long)(unsigned(x) * 998244351u) * mod()) >> 32;
	}
	[[gnu::always_inline]] Mint() = default;
	template <class I, std::enable_if_t<mint_integral, int> = 0>
	Mint(I x) {
		long long r = x % (long long)mod();
		if (r < 0) r += mod();
		v = reduce((unsigned long long)r * 932051910);
	}
	[[nodiscard, gnu::always_inline]] static Mint raw(unsigned x) {
		Mint a;
		a.v = reduce((unsigned long long)x * 932051910);
		return a;
	}
	[[nodiscard, gnu::always_inline]] static Mint word(unsigned x) {
		Mint a;
		a.v = x;
		return a;
	}
	[[nodiscard, gnu::always_inline]] unsigned val() const {
		unsigned r = reduce(v);
		return r < mod() ? r : r - mod();
	}
	friend std::istream &operator>>(std::istream &is, Mint &x) {
		long long value;
		if (is >> value) x = Mint(value);
		return is;
	}
	friend std::ostream &operator<<(std::ostream &os, const Mint &x) {
		return os << x.val();
	}
	[[gnu::always_inline]] Mint &operator+=(Mint b) {
		v += b.v;
		if (v >= 2 * mod()) v -= 2 * mod();
		return *this;
	}
	[[gnu::always_inline]] Mint &operator-=(Mint b) {
		v -= b.v;
		if (v >> 31) v += 2 * mod();
		return *this;
	}
	[[gnu::always_inline]] Mint &operator*=(Mint b) {
		v = reduce((unsigned long long)v * b.v);
		return *this;
	}
	Mint &operator/=(Mint b) { return *this *= b.inv(); }
	Mint &operator++() { return *this += Mint(1); }
	Mint &operator--() { return *this -= Mint(1); }
	Mint operator++(int) {
		Mint a = *this;
		++*this;
		return a;
	}
	Mint operator--(int) {
		Mint a = *this;
		--*this;
		return a;
	}
	[[nodiscard, gnu::always_inline]] Mint operator+() const { return *this; }
	[[nodiscard, gnu::always_inline]] Mint operator-() const { return word(v ? 2 * mod() - v : 0); }
	[[nodiscard, gnu::always_inline]] Mint operator+(Mint b) const { return Mint(*this) += b; }
	[[nodiscard, gnu::always_inline]] Mint operator-(Mint b) const { return Mint(*this) -= b; }
	[[nodiscard, gnu::always_inline]] Mint operator*(Mint b) const { return Mint(*this) *= b; }
	[[nodiscard]] Mint operator/(Mint b) const { return Mint(*this) /= b; }
	[[nodiscard, gnu::always_inline]] bool operator==(Mint b) const { return (v < mod() ? v : v - mod()) == (b.v < mod() ? b.v : b.v - mod()); }
	[[nodiscard, gnu::always_inline]] bool operator!=(Mint b) const { return !(*this == b); }
	template <class I, std::enable_if_t<mint_integral, int> = 0>
	[[nodiscard]] friend Mint operator+(I a, Mint b) { return Mint(a) + b; }
	template <class I, std::enable_if_t<mint_integral, int> = 0>
	[[nodiscard]] friend Mint operator-(I a, Mint b) { return Mint(a) - b; }
	template <class I, std::enable_if_t<mint_integral, int> = 0>
	[[nodiscard]] friend Mint operator*(I a, Mint b) { return Mint(a) * b; }
	template <class I, std::enable_if_t<mint_integral, int> = 0>
	[[nodiscard]] friend Mint operator/(I a, Mint b) { return Mint(a) / b; }
	template <class I, std::enable_if_t<mint_integral, int> = 0>
	[[nodiscard]] friend bool operator==(I a, Mint b) { return Mint(a) == b; }
	template <class I, std::enable_if_t<mint_integral, int> = 0>
	[[nodiscard]] friend bool operator!=(I a, Mint b) { return Mint(a) != b; }
	[[nodiscard]] Mint pow(long long n) const {
		assert(n >= 0);
		if (!n) return Mint(1);
		Mint r(1), a = *this;
		for (; n > 1; n >>= 1, a *= a)
			if (n & 1) r *= a;
		return r * a;
	}
	[[nodiscard, gnu::noinline]] Mint inv() const {
		assert(*this != Mint(0));
		Mint a2 = *this * *this, a3 = a2 * *this;
		Mint a7 = a3 * a3 * *this;
		Mint a31 = a7;
		for (int i = 0; i < 2; ++i) a31 *= a31;
		a31 *= a3;
		Mint r = a7;
		for (int i = 0; i < 3; ++i) r *= r;
		r *= a3;
		r *= r;
		for (int j = 0; j < 4; ++j) {
			for (int i = 0; i < 5; ++i) r *= r;
			r *= a31;
		}
		for (int i = 0; i < 3; ++i) r *= r;
		return r * a7;
	}
};
} // namespace mint_detail

using mint = mint_detail::Mint;
using fps_mint = mint;

#if defined(__GNUC__) && !defined(__clang__) && (defined(__x86_64__) || defined(__i386__))
#pragma GCC pop_options
#endif

#line 5 "components/mint_arithmetic.hpp"
#if defined(__x86_64__) || defined(__i386__)
#include <immintrin.h>
#endif

#if defined(__GNUC__) && !defined(__clang__) && (defined(__x86_64__) || defined(__i386__))
#pragma GCC push_options
#pragma GCC optimize("O3")
#endif

namespace mint_detail {
struct Ops {
	using T = mint;
	typedef unsigned int U __attribute__((may_alias));
	static constexpr U mod = 998244353, twice = 2 * mod, one = 301989884;
	static U reduce(unsigned long long x) { return T::reduce(x); }
	static U mul(U a, U b) { return reduce((unsigned long long)a * b); }
	static U add(U a, U b) {
		U c = a + b;
		return c < twice ? c : c - twice;
	}
	static U sub(U a, U b) {
		U c = a - b;
		return c < twice ? c : c + twice;
	}
	static U in(U a) { return mul(a, 932051910); }
	static U out(U a) {
		a = reduce(a);
		return a < mod ? a : a - mod;
	}
	static U quotient(U r) { return ((unsigned long long)r << 32) / mod; }
	static U fixed_mul(U a, U r, U q) {
		return a * r - U(((unsigned long long)a * q) >> 32) * mod;
	}

#if defined(__x86_64__) || defined(__i386__)
	static bool vectorized() {
#if defined(__AVX2__)
		return true;
#else
		static const bool available = __builtin_cpu_supports("avx2");
		return available;
#endif
	}
	using I8 = __m256i;
	__attribute__((target("avx2"))) static I8 load8(const U *a) {
		return _mm256_loadu_si256(reinterpret_cast<const I8 *>(a));
	}
	__attribute__((target("avx2"))) static void store8(U *a, I8 x) {
		_mm256_storeu_si256(reinterpret_cast<I8 *>(a), x);
	}
	__attribute__((target("avx2"), always_inline)) static inline I8 reduce8(I8 x, I8 y) {
		I8 r = _mm256_set1_epi32(998244351), p = _mm256_set1_epi32(mod);
		x = _mm256_add_epi64(x, _mm256_mul_epu32(_mm256_mul_epu32(x, r), p));
		y = _mm256_add_epi64(y, _mm256_mul_epu32(_mm256_mul_epu32(y, r), p));
		return _mm256_blend_epi32(_mm256_srli_epi64(x, 32), y, 0xaa);
	}
	__attribute__((target("avx2"))) static I8 mul8(I8 a, I8 b) {
		return reduce8(_mm256_mul_epu32(a, b), _mm256_mul_epu32(_mm256_srli_epi64(a, 32), _mm256_srli_epi64(b, 32)));
	}
	__attribute__((target("avx2"))) static I8 fixed_mul8(I8 a, I8 r, I8 q) {
		I8 x = _mm256_srli_epi64(_mm256_mul_epu32(a, q), 32);
		I8 y = _mm256_mul_epu32(_mm256_srli_epi64(a, 32), q);
		I8 t = _mm256_blend_epi32(x, y, 0xaa);
		return _mm256_sub_epi32(_mm256_mullo_epi32(a, r),
								_mm256_mullo_epi32(t, _mm256_set1_epi32(mod)));
	}
	__attribute__((target("avx2"))) static I8 fixed_even8(I8 a, I8 r, I8 q) {
		I8 t = _mm256_srli_epi64(_mm256_mul_epu32(a, q), 32);
		return _mm256_sub_epi64(_mm256_mul_epu32(a, r),
								_mm256_mul_epu32(t, _mm256_set1_epi32(mod)));
	}
	__attribute__((target("avx2"))) static I8 add8(I8 a, I8 b) {
		I8 x = _mm256_add_epi32(a, b);
		return _mm256_min_epu32(x, _mm256_sub_epi32(x, _mm256_set1_epi32(twice)));
	}
	__attribute__((target("avx2"))) static I8 sub8(I8 a, I8 b) {
		I8 x = _mm256_sub_epi32(a, b);
		return _mm256_min_epu32(x, _mm256_add_epi32(x, _mm256_set1_epi32(twice)));
	}

#endif
};
}

#if defined(__GNUC__) && !defined(__clang__) && (defined(__x86_64__) || defined(__i386__))
#pragma GCC pop_options
#endif

#line 10 "components/fps_output.hpp"
#include <vector>

#if defined(__linux__) && defined(__x86_64__) && defined(__GNUC__) && !defined(__clang__)
#include <immintrin.h>
#include <sys/mman.h>
#pragma GCC push_options
_Pragma("GCC target(\"avx2\")")
#pragma GCC optimize("O3")

namespace fps_detail {

inline void prepare_fps_output(std::vector<mint> &out, std::size_t n) {
	if (n <= out.capacity()) return;
	out.reserve(n);
	if (n - out.size() < (1u << 17)) return;
	const auto first = reinterpret_cast<std::uintptr_t>(out.data() + out.size());
	const auto end = reinterpret_cast<std::uintptr_t>(out.data() + n);
	[[maybe_unused]] const auto advise = [first, end](std::uintptr_t alignment, int advice) {
		const auto lo = (first + alignment - 1) & ~(alignment - 1), hi = end & ~(alignment - 1);
		if (lo < hi) static_cast<void>(madvise(reinterpret_cast<void *>(lo), hi - lo, advice));
	};
#if defined(MADV_HUGEPAGE)
	advise(2u * 1024 * 1024, MADV_HUGEPAGE);
#endif
#if defined(MADV_POPULATE_WRITE)
	advise(4096, MADV_POPULATE_WRITE);
#endif
}

template <bool montgomery>
inline void append_fps_words(std::vector<mint> &out, const unsigned *in, std::size_t n) {
	static_assert(sizeof(mint) == sizeof(unsigned) && std::is_trivially_copyable_v<mint>);
	if (n == 0) return;
	if (n > out.max_size() - out.size()) throw std::length_error("fps output");
	prepare_fps_output(out, out.size() + n);
	std::array<mint, 64> batch;
	std::size_t i = 0;
	for (; i + 64 <= n; i += 64) {
		if constexpr (montgomery) std::memcpy(static_cast<void *>(batch.data()), in + i, sizeof(batch));
		else {
#pragma GCC unroll 8
			for (int j = 0; j < 64; j += 8) {
				const auto x = mint_detail::Ops::mul8(_mm256_loadu_si256(reinterpret_cast<const __m256i *>(in + i + j)), _mm256_set1_epi32(932051910u));
				std::memcpy(static_cast<void *>(batch.data() + j), &x, sizeof(x));
			}
		}
		out.insert(out.end(), batch.begin(), batch.end());
	}
	for (; i < n; ++i) {
		if constexpr (montgomery) out.emplace_back(mint::word(in[i]));
		else out.emplace_back(mint::raw(in[i]));
	}
}
}
#pragma GCC pop_options
#endif

#line 1 "components/fps_write.hpp"

#line 10 "components/fps_write.hpp"

namespace detail {

struct alignas(32) Lanes {
	std::uint32_t lane[8];
};

constexpr Lanes splat(std::uint32_t x) { return {{x, x, x, x, x, x, x, x}}; }
constexpr Lanes all16(std::uint32_t x) { return splat(x * 0x10001); }

constexpr int Z = 0x80;

constexpr Lanes shuffle(const int (&b)[16]) {
	Lanes t{};
	for (int i = 0; i < 32; ++i) t.lane[i / 4] |= std::uint32_t(b[i % 16] & 0xFF) << 8 * (i % 4);
	return t;
}

struct Constants {
	Lanes div10 = splat(429496730);
	Lanes div10e5 = splat(1407374884);
	Lanes pack = splat(1 - 10000 * 65536u);
	Lanes ten = splat(10);
	Lanes tail = splat(' ' << 8 | '0');
	Lanes by100 = all16(5243), hundred = all16(100);
	Lanes by10 = all16(6554);
	Lanes tens_out = all16(2559);
	Lanes ones = splat(~0u), zero = splat(0x30303030), space = splat(0x20202020);

Lanes chunk0[2] = {shuffle({0, 1, 2, 3, 4, 5, 6, 7, Z, Z, 8, 9, 10, 11, 12, 13}),
					shuffle({Z, Z, Z, Z, Z, Z, Z, Z, 0, 1, Z, Z, Z, Z, Z, Z})};
	Lanes chunk1[3] = {shuffle({14, 15, Z, Z, Z, Z, Z, Z, Z, Z, Z, Z, Z, Z, Z, Z}),
					shuffle({Z, Z, 4, 5, Z, Z, Z, Z, Z, Z, Z, Z, 8, 9, Z, Z}),
					shuffle({Z, Z, Z, Z, 0, 1, 2, 3, 4, 5, 6, 7, Z, Z, 8, 9})};
	Lanes chunk2[3] = {shuffle({10, 11, 12, 13, 14, 15, Z, Z, Z, Z, Z, Z, Z, Z, Z, Z}),
					shuffle({Z, Z, Z, Z, Z, Z, 12, 13, Z, Z, Z, Z, Z, Z, Z, Z}),
					shuffle({Z, Z, Z, Z, Z, Z, Z, Z, 0, 1, 2, 3, 4, 5, 6, 7})};
	Lanes chunk3[3] = {shuffle({0, 1, Z, Z, Z, Z, Z, Z, Z, Z, 4, 5, Z, Z, Z, Z}),
					shuffle({Z, Z, 8, 9, 10, 11, 12, 13, 14, 15, Z, Z, Z, Z, Z, Z}),
					shuffle({Z, Z, Z, Z, Z, Z, Z, Z, Z, Z, Z, Z, 0, 1, 2, 3})};
	Lanes chunk4[2] = {shuffle({4, 5, 6, 7, Z, Z, 8, 9, 10, 11, 12, 13, 14, 15, Z, Z}),
					shuffle({Z, Z, Z, Z, 8, 9, Z, Z, Z, Z, Z, Z, Z, Z, 12, 13})};
};

inline constexpr Constants kConstants{};

[[gnu::always_inline]] inline __m256i load(const Lanes& l) {
	return _mm256_load_si256(reinterpret_cast<const __m256i*>(&l));
}

[[gnu::always_inline]] inline __m256i shuffle(__m256i x, const Lanes& control) {
	return _mm256_shuffle_epi8(x, load(control));
}

struct Divided {
	__m256i halves, tail;
};

[[gnu::always_inline]] inline Divided divide(__m256i v, __m256i odd, const Constants& k) {
	const __m256i w = _mm256_blend_epi32(_mm256_srli_epi64(_mm256_mul_epu32(v, load(k.div10)), 32),
										_mm256_mul_epu32(odd, load(k.div10)), 0xAA);
	const __m256i h = _mm256_blend_epi32(_mm256_srli_epi64(_mm256_mul_epu32(v, load(k.div10e5)), 47),
										_mm256_srli_epi64(_mm256_mul_epu32(odd, load(k.div10e5)), 15), 0xAA);
	const __m256i halves = _mm256_add_epi32(_mm256_slli_epi32(w, 16), _mm256_mullo_epi32(h, load(k.pack)));
	const __m256i tail = _mm256_sub_epi32(_mm256_add_epi32(v, load(k.tail)), _mm256_mullo_epi32(w, load(k.ten)));
	return {halves, tail};
}

[[gnu::always_inline]] inline __m256i two_digits(__m256i z, const Constants& k) {
	const __m256i tens = _mm256_mulhi_epu16(z, load(k.by10));
	return _mm256_sub_epi16(_mm256_slli_epi16(z, 8), _mm256_mullo_epi16(tens, load(k.tens_out)));
}

[[gnu::always_inline]] inline __m256i text(__m256i digits, const Constants& k) {
	const __m256i upto = _mm256_xor_si256(digits, _mm256_add_epi64(digits, load(k.ones)));
	return _mm256_blendv_epi8(_mm256_or_si256(digits, load(k.zero)), load(k.space), upto);
}

[[gnu::always_inline]] inline void digits(const Divided& d, __m256i& lo, __m256i& hi, const Constants& k) {
	const __m256i hundreds = _mm256_srli_epi16(_mm256_mulhi_epu16(d.halves, load(k.by100)), 3);
	const __m256i rest = _mm256_sub_epi16(d.halves, _mm256_mullo_epi16(hundreds, load(k.hundred)));
	lo = text(two_digits(_mm256_unpacklo_epi16(hundreds, rest), k), k);
	hi = text(two_digits(_mm256_unpackhi_epi16(hundreds, rest), k), k);
}

[[gnu::always_inline]] inline void store(char* p, const Divided& lo, const Divided& hi, const Constants& k) {
	__m256i d01, d23, d45, d67;
	const auto either = [&](__m256i a, __m256i b) __attribute__((target("avx2"))) { return _mm256_or_si256(a, b); };
	digits(lo, d01, d23, k);
	digits(hi, d45, d67, k);
	const __m256i c0 = either(shuffle(d01, k.chunk0[0]), shuffle(lo.tail, k.chunk0[1]));
	const __m256i c1 = either(either(shuffle(d01, k.chunk1[0]), shuffle(lo.tail, k.chunk1[1])), shuffle(d23, k.chunk1[2]));
	const __m256i c2 = either(either(shuffle(d23, k.chunk2[0]), shuffle(lo.tail, k.chunk2[1])), shuffle(d45, k.chunk2[2]));
	const __m256i c3 = either(either(shuffle(hi.tail, k.chunk3[0]), shuffle(d45, k.chunk3[1])), shuffle(d67, k.chunk3[2]));
	const __m256i c4 = either(shuffle(d67, k.chunk4[0]), shuffle(hi.tail, k.chunk4[1]));

const auto whole = [p](int c, __m256i chunk) __attribute__((target("avx2"))) { _mm256_storeu_si256(reinterpret_cast<__m256i*>(p + 64 + 16 * c), chunk); };
	const auto low = [p](int c, __m256i chunk) __attribute__((target("avx2"))) { _mm_store_si128(reinterpret_cast<__m128i*>(p + 16 * c), _mm256_castsi256_si128(chunk)); };
	whole(4, c4), whole(3, c3), whole(2, c2), whole(1, c1), whole(0, c0);
	low(0, c0), low(1, c1), low(2, c2), low(3, c3), low(4, c4);
}

[[gnu::always_inline]] inline void divide16(const std::uint32_t* x, Divided& lo, Divided& hi, const Constants& k) {
	const auto at = [x](int i) __attribute__((target("avx2"))) { return _mm256_loadu_si256(reinterpret_cast<const __m256i*>(x + i)); };
	const auto both = [x](int i) __attribute__((target("avx2"))) { return _mm256_broadcastsi128_si256(_mm_loadu_si128(reinterpret_cast<const __m128i*>(x + i))); };
	const __m256i l = _mm256_blend_epi32(at(0), both(8), 0xF0), h = _mm256_blend_epi32(at(8), both(4), 0x0F);
	lo = divide(l, _mm256_srli_epi64(l, 32), k);
	hi = divide(h, _mm256_srli_epi64(h, 32), k);
}

inline void format(char* p, const std::uint32_t* x, std::size_t count, const Constants& k) {
	Divided dl, dh;
	divide16(x, dl, dh, k);
	for (std::size_t i = 16; i < count; i += 16, p += 160) {
		Divided nl, nh;
		divide16(x + i, nl, nh, k);
		store(p, dl, dh, k);
		dl = nl, dh = nh;
	}
	store(p, dl, dh, k);
}

}

inline constexpr std::size_t kBlock = 25600;
inline constexpr std::size_t kTextBytes = 10 * kBlock;

}

namespace fps_detail {
[[gnu::always_inline]] inline __m256i io_values(__m256i x) {
	const auto p = _mm256_set1_epi32(mint::mod()), ni = _mm256_set1_epi32(998244351);
	auto even = _mm256_and_si256(x, _mm256_set1_epi64x(0xffffffffULL));
	auto odd = _mm256_srli_epi64(x, 32);
	even = _mm256_add_epi64(even, _mm256_mul_epu32(_mm256_mul_epu32(even, ni), p));
	odd = _mm256_add_epi64(odd, _mm256_mul_epu32(_mm256_mul_epu32(odd, ni), p));
	const auto value = _mm256_blend_epi32(_mm256_srli_epi64(even, 32), odd, 0xaa);
	return _mm256_min_epu32(value, _mm256_sub_epi32(value, p));
}

inline void write_coefficients_native(FastOutput &output, const mint *in, std::size_t n) {
	alignas(32) unsigned values[25600];
	alignas(32) char text[256000];
	const auto &k = fps_io_fields::detail::kConstants;
	for (std::size_t i = 0; i < n; i += 25600) {
		const auto count = std::min<std::size_t>(25600, n - i), full = count / 16 * 16;
		char *end = text;
		static constexpr char zeros[] = "0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ";
		const auto empty = [&]() __attribute__((target("avx2"))) {
			std::size_t at = 0;
			const auto p = _mm256_set1_epi32(mint::mod());
			for (; at + 8 <= count; at += 8) {
				__m256i x;
				std::memcpy(&x, in + i + at, sizeof(x));
				x = _mm256_min_epu32(x, _mm256_sub_epi32(x, p));
				if (!_mm256_testz_si256(x, x)) return false;
			}
			for (; at < count; ++at) if (in[i + at] != mint(0)) return false;
			return true;
		};
		if (empty()) {
			for (std::size_t at = 0; at < count; at += 16) std::memcpy(text + 2 * at, zeros, 32);
			end += 2 * count;
		} else {
			std::size_t j = 0;
			for (; j + 8 <= count; j += 8) {
				alignas(32) unsigned words[8];
				std::memcpy(words, in + i + j, sizeof(words));
				_mm256_storeu_si256(reinterpret_cast<__m256i *>(values + j), io_values(_mm256_load_si256(reinterpret_cast<const __m256i *>(words))));
			}
			for (; j < count; ++j) values[j] = in[i + j].val();
			auto zero = [&](std::size_t at) __attribute__((target("avx2"))) {
				auto x = _mm256_or_si256(_mm256_loadu_si256(reinterpret_cast<const __m256i *>(values + at)), _mm256_loadu_si256(reinterpret_cast<const __m256i *>(values + at + 8)));
				return _mm256_testz_si256(x, x);
			};
			std::size_t at = 0;
			while (at < full) {
				if (zero(at)) { std::memcpy(end, zeros, 32); end += 32; at += 16; }
				else {
					const auto first = at;
					do at += 16; while (at < full && !zero(at));
					fps_io_fields::detail::format(end, values + first, at - first, k);
					end += 10 * (at - first);
				}
			}
			if (full < count) {
				alignas(32) unsigned rest[16] = {};
				alignas(16) char last[160];
				std::copy_n(values + full, count - full, rest);
				bool empty = true;
				for (auto x : rest) empty &= x == 0;
				if (empty) { std::memcpy(end, zeros, 2 * (count - full)); end += 2 * (count - full); }
				else {
					fps_io_fields::detail::format(last, rest, 16, k);
					std::memcpy(end, last, 10 * (count - full));
					end += 10 * (count - full);
				}
			}
		}
		if (i + count == n) end[-1] = '\n';
		fast_io_detail::BulkOutput::write(output, std::string_view(text, end - text));
	}
	if (!n) output.write('\n');
}
}
#pragma GCC pop_options
#endif

namespace fps_detail {
inline void write_coefficients(FastOutput &output, const mint *in, std::size_t n) {
#if defined(__linux__) && defined(__x86_64__) && defined(__GNUC__) && !defined(__clang__)
	if (__builtin_cpu_supports("avx2")) return write_coefficients_native(output, in, n);
#endif
	for (std::size_t i = 0; i < n; ++i) { output.write(in[i].val()); output.write(' '); }
	output.write('\n');
}
}

#line 8 "components/fps_io.hpp"

#if defined(__linux__) && defined(__x86_64__) && defined(__GNUC__) && !defined(__clang__)
#include <immintrin.h>
#pragma GCC push_options
_Pragma("GCC target(\"avx2\")")
#pragma GCC optimize("O3")

namespace fps_detail {
inline void read_coefficients(FastInput &input, std::vector<mint> &out, std::size_t n) {
	if (!n) return;
	if (n > out.max_size() - out.size()) throw std::length_error("fps input");
	prepare_fps_output(out, out.size() + n);
	fast_io_detail::BulkInput::read_blocks(input, n, [&](const unsigned *values, std::size_t count) {
		append_fps_words<false>(out, values, count);
	});
}
}
#pragma GCC pop_options
#endif

#ifndef CP_COMPONENTS_FPS_IO_HPP
#define CP_COMPONENTS_FPS_IO_HPP

#include "fast_io_bulk.hpp"
#include "fps_output.hpp"
#include "fps_write.hpp"
#include <vector>

#if defined(__linux__) && defined(__x86_64__) && defined(__GNUC__) && !defined(__clang__)
#include <immintrin.h>
#pragma GCC push_options
_Pragma("GCC target(\"avx2\")")
#pragma GCC optimize("O3")

namespace fps_detail {
inline void read_coefficients(FastInput &input, std::vector<mint> &out, std::size_t n) {
	if (!n) return;
	if (n > out.max_size() - out.size()) throw std::length_error("fps input");
	prepare_fps_output(out, out.size() + n);
	fast_io_detail::BulkInput::read_blocks(input, n, [&](const unsigned *values, std::size_t count) {
		append_fps_words<false>(out, values, count);
	});
}
}
#pragma GCC pop_options
#endif
#endif

#line 1 "components/fps_io.hpp"



#line 1 "components/fast_io_bulk.hpp"



#line 1 "components/fast_io_output.hpp"



#line 1 "components/fast_io.hpp"



#include <algorithm>
#include <array>
#include <bit>
#include <cassert>
#include <charconv>
#include <cstddef>
#include <cstdint>
#include <cstdio>
#include <cstring>
#include <memory>
#include <stdexcept>
#include <string>
#include <string_view>
#include <type_traits>
#if defined(__unix__) || defined(__APPLE__)
#include <sys/mman.h>
#include <sys/stat.h>
#include <unistd.h>
#endif

namespace fast_io_detail { struct BulkInput; struct BulkOutput; }

class FastInput {
	friend struct fast_io_detail::BulkInput;
	static constexpr std::size_t capacity = 1 << 16;
	FILE *file = nullptr;
	std::unique_ptr<char[]> buffer;
	const char *current = "", *end = current;
	std::string scratch;
	void *mapping = nullptr;
	std::size_t mapped_size = 0;
	bool refill() {
		if (!file) return false;
		std::size_t n = std::fread(buffer.get(), 1, capacity, file);
		current = buffer.get();
		end = current + n;
		return n;
	}
	static const char *scan(const char *p, const char *end) {
		while (end - p >= 8) {
			std::uint64_t x;
			std::memcpy(&x, p, 8);
			if ((x - 0x2121212121212121ULL) & ~x & 0x8080808080808080ULL) break;
			p += 8;
		}
		while (p != end && static_cast<unsigned char>(*p) > ' ') ++p;
		return p;
	}

public:
	explicit FastInput(FILE *input = stdin) {
#if defined(__unix__) || defined(__APPLE__)
		struct stat status;
		int fd = fileno(input);
		auto offset = ftello(input);
		if (fd >= 0 && offset >= 0 && !fstat(fd, &status) && S_ISREG(status.st_mode) && status.st_size > offset) {
			mapped_size = status.st_size;
			mapping = mmap(nullptr, mapped_size, PROT_READ, MAP_PRIVATE, fd, 0);
			if (mapping != MAP_FAILED) {
				current = static_cast<const char *>(mapping) + offset;
				end = static_cast<const char *>(mapping) + mapped_size;
				return;
			}
			mapping = nullptr;
		}
#endif
		file = input;
		buffer = std::make_unique_for_overwrite<char[]>(capacity);
	}
	explicit FastInput(std::string_view input) : current(input.empty() ? "" : input.data()), end(current + input.size()) {}
	FastInput(const FastInput &) = delete;
	FastInput &operator=(const FastInput &) = delete;
	~FastInput() {
#if defined(__unix__) || defined(__APPLE__)
		if (mapping) munmap(mapping, mapped_size);
#endif
	}
	std::string_view token() {
		while (true) {
			while (current != end && static_cast<unsigned char>(*current) <= ' ') ++current;
			if (current != end) break;
			if (!refill()) return {};
		}
		const char *begin = current;
		current = scan(current, end);
		if (current != end || !file) return {begin, std::size_t(current - begin)};
		scratch.assign(begin, end);
		while (refill()) {
			current = scan(current, end);
			scratch.append(buffer.get(), std::size_t(current - buffer.get()));
			if (current != end) break;
		}
		return scratch;
	}
	template <class T>
	bool read(T &value) {
		if constexpr (std::is_integral_v<T> && !std::is_same_v<T, bool>) {
			while (true) {
				while (current != end && static_cast<unsigned char>(*current) <= ' ') ++current;
				if (current != end) break;
				if (!refill()) return false;
			}
			if constexpr (std::is_unsigned_v<T> && sizeof(T) == 4 && std::endian::native == std::endian::little) {
				if (end - current >= 11) {
					std::uint64_t x;
					std::memcpy(&x, current, 8);
					if (!(((x + 0x4646464646464646ULL) | (x - 0x3030303030303030ULL)) & 0x8080808080808080ULL)) {
						x &= 0x0f0f0f0f0f0f0f0fULL;
						x = (x * 10 + (x >> 8)) & 0x00ff00ff00ff00ffULL;
						x = (x * 100 + (x >> 16)) & 0x0000ffff0000ffffULL;
						std::uint64_t v = std::uint32_t(x * 10000 + (x >> 32));
						int n = 8;
						for (; n < 10 && static_cast<unsigned char>(current[n]) - unsigned('0') < 10; ++n)
							v = v * 10 + (current[n] - '0');
						if (static_cast<unsigned char>(current[n]) <= ' ' && v <= 0xffffffffULL) {
							current += n;
							value = T(v);
							return true;
						}
					}
				}
			}
			T parsed{};
			auto result = std::from_chars(current, end, parsed);
			if (result.ec == std::errc{} && (result.ptr != end ? static_cast<unsigned char>(*result.ptr) <= ' ' : !file)) {
				current = result.ptr;
				value = parsed;
				return true;
			}
		}
		auto s = token();
		if (s.empty()) return false;
		if constexpr (std::is_same_v<T, bool>) {
			assert(s == "0" || s == "1");
			value = s[0] == '1';
		} else if constexpr (std::is_integral_v<T>) {
			auto result = std::from_chars(s.data(), s.data() + s.size(), value);
			assert(result.ec == std::errc{} && result.ptr == s.data() + s.size());
			(void)result;
		} else value = T(s);
		return true;
	}
};

class FastOutput {
	friend struct fast_io_detail::BulkOutput;
	inline static constexpr auto quads = [] {
		std::array<std::array<char, 4>, 10000> table{};
		for (int i = 0; i < 10000; ++i)
			table[i] = {char('0' + i / 1000), char('0' + i / 100 % 10), char('0' + i / 10 % 10), char('0' + i % 10)};
		return table;
	}();
	FILE *file;
	std::unique_ptr<char[]> buffer = std::make_unique_for_overwrite<char[]>(1 << 16);
	std::size_t capacity = 1 << 16, length = 0;
	char *reserve(std::size_t n) {
		if (n > capacity - length) {
			if (!flush()) throw std::runtime_error("output failed");
			if (n > capacity) {
				std::size_t next = std::max(n, 2 * capacity);
				buffer = std::make_unique_for_overwrite<char[]>(next);
				capacity = next;
			}
		}
		return buffer.get() + length;
	}

public:
	explicit FastOutput(FILE *output = stdout) : file(output) {}
	FastOutput(const FastOutput &) = delete;
	FastOutput &operator=(const FastOutput &) = delete;
	~FastOutput() { flush(); }
	bool flush() {
		std::size_t offset = 0;
		while (offset < length) {
			auto n = std::fwrite(buffer.get() + offset, 1, length - offset, file);
			if (!n) {
				std::memmove(buffer.get(), buffer.get() + offset, length - offset);
				length -= offset;
				return false;
			}
			offset += n;
		}
		length = 0;
		return std::fflush(file) == 0;
	}
	void write(char c) { *reserve(1) = c; ++length; }
	void write(__uint128_t value) {
		char digits[40];
		char *end = digits + sizeof(digits), *begin = end;
		do { *--begin = char('0' + value % 10); value /= 10; } while (value);
		write(std::string_view(begin, end - begin));
	}
	void write(__int128_t value) {
		__uint128_t magnitude = __uint128_t(value);
		if (value < 0) write('-'), magnitude = 0 - magnitude;
		write(magnitude);
	}
	void write(std::string_view s) {
		char *out = reserve(s.size());
		if (!s.empty()) std::memcpy(out, s.data(), s.size());
		length += s.size();
	}
	template <class T> requires std::is_integral_v<T>
	void write(T value) {
		char *out = reserve(32);
		if constexpr (std::is_same_v<T, bool>) *out = '0' + value, ++length;
		else if constexpr (std::is_unsigned_v<T> && sizeof(T) == 4) {
			char *p = out;
			if (value >= 100000000) {
				auto high = value / 100000000, low = value % 100000000;
				p = std::to_chars(p, p + 2, high).ptr;
				std::memcpy(p, quads[low / 10000].data(), 4);
				std::memcpy(p + 4, quads[low % 10000].data(), 4);
				p += 8;
			} else if (value >= 10000) {
				p = std::to_chars(p, p + 4, value / 10000).ptr;
				std::memcpy(p, quads[value % 10000].data(), 4);
				p += 4;
			} else p = std::to_chars(p, p + 4, value).ptr;
			length += p - out;
		} else length += std::to_chars(out, out + 32, value).ptr - out;
	}
	template <class T> requires requires(const T &value, char *out) { value.decimal_capacity(); value.to_chars(out); }
	void write(const T &value) {
		char *out = reserve(value.decimal_capacity());
		length += value.to_chars(out) - out;
	}
	template <class T>
	void println(const T &value) { write(value); write('\n'); }
};


#line 5 "components/fast_io_output.hpp"
#include <limits>
#include <utility>

namespace fast_io_detail {
struct BulkOutput {
	static char *small(char *out, std::uint32_t value) {
		unsigned n = 1 + (value >= 10) + (value >= 100) + (value >= 1000);
		if constexpr (std::endian::native == std::endian::little) {
			std::uint32_t word;
			std::memcpy(&word, FastOutput::quads[value].data(), 4);
			word >>= (4 - n) * 8;
			std::memcpy(out, &word, 4);
		} else std::memcpy(out, FastOutput::quads[value].data() + 4 - n, n);
		return out + n;
	}
	static char *number(char *out, std::uint32_t value) {
		if (value >= 100000000) {
			out = small(out, value / 100000000);
			value %= 100000000;
			std::memcpy(out, FastOutput::quads[value / 10000].data(), 4);
			std::memcpy(out + 4, FastOutput::quads[value % 10000].data(), 4);
			return out + 8;
		}
		if (value >= 10000) {
			out = small(out, value / 10000);
			std::memcpy(out, FastOutput::quads[value % 10000].data(), 4);
			return out + 4;
		}
		return small(out, value);
	}
	template<std::size_t Fields, class Read>
#if defined(__GNUC__) || defined(__clang__)
	[[gnu::noinline]]
#endif
	static void records(FastOutput &out, std::size_t count, Read read) {
		constexpr auto bytes = 11 * Fields;
		constexpr auto block = std::max<std::size_t>(1, (1 << 16) / bytes);
		for (std::size_t base = 0; base < count;) {
			const auto n = std::min(count - base, block);
			char *begin = out.reserve(n * bytes), *p = begin;
			for (std::size_t i = 0; i < n; ++i) {
				auto values = read(base + i);
				for (std::size_t j = 0; j < Fields; ++j) {
					p = number(p, std::uint32_t(values[j]));
					*p++ = j + 1 == Fields ? '\n' : ' ';
				}
			}
			out.length += p - begin;
			base += n;
		}
	}
	static void write(FastOutput &out, std::string_view text) {
		if (text.size() <= (1 << 16)) return out.write(text);
		if (out.length && !out.flush()) throw std::runtime_error("output failed");
		std::size_t done = 0;
		while (done < text.size()) {
			const auto written = std::fwrite(text.data() + done, 1, text.size() - done, out.file);
			if (!written) throw std::runtime_error("output failed");
			done += written;
		}
	}
};

}

template<std::size_t Fields, class Read>
void write_unsigned_records(FastOutput &output, std::size_t count, Read read) {
	static_assert(Fields > 0 && Fields <= std::numeric_limits<std::size_t>::max() / 11);
	fast_io_detail::BulkOutput::records<Fields>(output, count, std::move(read));
}


#line 6 "components/fast_io_bulk.hpp"
#include <span>
#line 8 "components/fast_io_bulk.hpp"

#if defined(__linux__) && defined(__x86_64__) && defined(__GNUC__) && !defined(__clang__)
#include <immintrin.h>
#pragma GCC push_options
_Pragma("GCC target(\"avx2\")")
#pragma GCC optimize("O3")
namespace fast_io_detail {
struct BulkInput {
	alignas(32) static inline constexpr auto rows = [] {
		std::array<std::array<unsigned char, 16>, 11> r{};
		for (int n = 0; n <= 10; ++n)
			for (int i = 0; i < 16; ++i) r[n][i] = i < 16 - n ? 128 : i - (16 - n);
		return r;
	}();
	static constexpr std::size_t kChunk = 1 << 17, kCapacity = kChunk / 4 + 66;

	static unsigned separators(__m256i bytes) {
		return unsigned(_mm256_movemask_epi8(_mm256_cmpeq_epi8(_mm256_max_epu8(bytes, _mm256_set1_epi8(' ')), _mm256_set1_epi8(' '))));
	}

	[[gnu::always_inline]] static __m256i two(const char *&p, __m256i &invalid) {
		const auto bytes = _mm256_loadu_si256(reinterpret_cast<const __m256i *>(p));
		const unsigned sep = separators(bytes), a = std::countr_zero(sep), b = std::countr_zero(sep & (sep - 1));
		if (a < 1 || a > 10 || b <= a + 1 || b - a - 1 > 10) [[unlikely]] {
			invalid = _mm256_set1_epi32(-1);
			p += std::min(b + 1, 32u);
			return _mm256_setzero_si256();
		}
		const auto window = _mm256_inserti128_si256(bytes, _mm_loadu_si128(reinterpret_cast<const __m128i *>(p + a + 1)), 1);
		const auto control = _mm256_set_m128i(_mm_loadu_si128(reinterpret_cast<const __m128i *>(rows[b - a - 1].data())),
			_mm_loadu_si128(reinterpret_cast<const __m128i *>(rows[a].data())));
		const auto digits = _mm256_shuffle_epi8(_mm256_sub_epi8(window, _mm256_set1_epi8('0')), control);
		invalid = _mm256_or_si256(invalid, _mm256_subs_epu8(digits, _mm256_set1_epi8(9)));
		p += b + 1;
		return _mm256_madd_epi16(_mm256_maddubs_epi16(digits, _mm256_set1_epi16(0x010a)), _mm256_set1_epi32(0x00010064));
	}

	static std::uint32_t one(const char *&p, bool &bad) {
		const auto bytes = _mm_loadu_si128(reinterpret_cast<const __m128i *>(p));
		const auto spaces = _mm_cmpeq_epi8(_mm_max_epu8(bytes, _mm_set1_epi8(' ')), _mm_set1_epi8(' '));
		const unsigned n = std::countr_zero(unsigned(_mm_movemask_epi8(spaces)) | 0x10000u);
		if (n < 1 || n > 10) [[unlikely]] {
			bad = true;
			p += std::min(n + 1, 16u);
			return 0;
		}
		const auto digits = _mm_shuffle_epi8(_mm_sub_epi8(bytes, _mm_set1_epi8('0')),
			_mm_loadu_si128(reinterpret_cast<const __m128i *>(rows[n].data())));
		const auto invalid = _mm_subs_epu8(digits, _mm_set1_epi8(9));
		bad |= !_mm_testz_si128(invalid, invalid);
		const auto groups = _mm_madd_epi16(_mm_maddubs_epi16(digits, _mm_set1_epi16(0x010a)), _mm_set1_epi32(0x00010064));
		const auto halves = _mm_madd_epi16(_mm_packus_epi32(groups, groups), _mm_set1_epi32(0x00012710));
		const std::uint64_t value = std::uint64_t(unsigned(_mm_cvtsi128_si32(halves))) * 100000000 + unsigned(_mm_extract_epi32(halves, 1));
		bad |= value > 0xffffffffULL;
		p += n + 1;
		return std::uint32_t(value);
	}

	static const char *after_separator(const char *p) {
		const unsigned mask = separators(_mm256_loadu_si256(reinterpret_cast<const __m256i *>(p)));
		return mask ? p + std::countr_zero(mask) + 1 : nullptr;
	}

	static std::size_t lockstep(const char *(&streams)[4], const char *const (&end)[4], unsigned *const (&out)[4], __m256i &invalid) {
		const char *p[4] = {streams[0], streams[1], streams[2], streams[3]};
		std::size_t done = 0;
		for (;;) {
			std::ptrdiff_t left = end[0] - p[0];
			for (int k = 1; k < 4; ++k) left = std::min(left, end[k] - p[k]);
			const auto steps = std::min(left / 33, std::ptrdiff_t((kCapacity - done) / 2));
			if (steps <= 0) break;
			for (auto j = steps; j; --j, done += 2) {
				const auto g0 = two(p[0], invalid), g1 = two(p[1], invalid), g2 = two(p[2], invalid), g3 = two(p[3], invalid);
				const auto k = _mm256_set1_epi32(0x00012710);
				const auto h01 = _mm256_castsi256_ps(_mm256_madd_epi16(_mm256_packus_epi32(g0, g1), k));
				const auto h23 = _mm256_castsi256_ps(_mm256_madd_epi16(_mm256_packus_epi32(g2, g3), k));
				const auto high = _mm256_castps_si256(_mm256_shuffle_ps(h01, h23, 0x88));
				const auto low = _mm256_castps_si256(_mm256_shuffle_ps(h01, h23, 0xdd));
				const auto overflow = _mm256_or_si256(_mm256_cmpgt_epi32(high, _mm256_set1_epi32(42)),
					_mm256_and_si256(_mm256_cmpeq_epi32(high, _mm256_set1_epi32(42)), _mm256_cmpgt_epi32(low, _mm256_set1_epi32(94967295))));
				invalid = _mm256_or_si256(invalid, overflow);
				const auto value = _mm256_add_epi32(_mm256_mullo_epi32(high, _mm256_set1_epi32(100000000)), low);
				const auto q = _mm256_permutevar8x32_epi32(value, _mm256_setr_epi32(0, 4, 1, 5, 2, 6, 3, 7));
				const auto q01 = _mm256_castsi256_si128(q), q23 = _mm256_extracti128_si256(q, 1);
				_mm_storel_epi64(reinterpret_cast<__m128i *>(out[0] + done), q01);
				_mm_storeh_pd(reinterpret_cast<double *>(out[1] + done), _mm_castsi128_pd(q01));
				_mm_storel_epi64(reinterpret_cast<__m128i *>(out[2] + done), q23);
				_mm_storeh_pd(reinterpret_cast<double *>(out[3] + done), _mm_castsi128_pd(q23));
			}
		}
		for (int k = 0; k < 4; ++k) streams[k] = p[k];
		return done;
	}

	template <class Emit>
	static std::size_t chunk_blocks(const char *&p, const char *limit, std::size_t count, unsigned *const (&out)[4], Emit &&emit) {
		const auto bytes = std::min({kChunk, 2 * (count - 32), std::size_t(limit - p) - 64}) & ~std::size_t(3);
		const std::size_t stream = bytes / 4;
		const char *p1 = after_separator(p + stream), *p2 = after_separator(p + 2 * stream);
		const char *p3 = after_separator(p + 3 * stream), *last = after_separator(p + bytes);
		if (!p1 || !p2 || !p3 || !last) return 0;
		const char *s[4] = {p, p1, p2, p3}, *const end[4] = {p1, p2, p3, last};
		auto invalid = _mm256_setzero_si256();
		const auto done = lockstep(s, end, out, invalid);
		bool bad = !_mm256_testz_si256(invalid, invalid);
		std::size_t n[4];
		for (int k = 0; k < 4; ++k) {
			n[k] = done;
			while (s[k] < end[k] && n[k] < kCapacity) out[k][n[k]++] = one(s[k], bad);
			bad |= s[k] != end[k];
		}
		const auto parsed = n[0] + n[1] + n[2] + n[3];
		if (bad || parsed > count) return 0;
		for (int k = 0; k < 4; ++k) emit(out[k], n[k]);
		p = last;
		return parsed;
	}

	static std::size_t chunk(const char *&p, const char *limit, unsigned *dst, std::size_t count, unsigned *scratch) {
		unsigned *const out[4] = {dst, scratch, scratch + kCapacity, scratch + 2 * kCapacity};
		bool first = true;
		unsigned *next = dst;
		return chunk_blocks(p, limit, count, out, [&](const unsigned *values, std::size_t n) {
			if (!first) std::memcpy(next, values, n * sizeof(unsigned));
			first = false;
			next += n;
		});
	}

	template <class Emit>
	static void read_blocks(FastInput &in, std::size_t count, Emit emit) {
		std::unique_ptr<unsigned[]> scratch;
		std::size_t done = 0;
		bool chunks = true;
		while (done < count) {
			const char *p = in.current;
			while (p != in.end && static_cast<unsigned char>(*p) <= ' ') ++p;
			while (chunks && count - done >= 1024 && in.end - p >= 1088) {
				if (!scratch) scratch = std::make_unique_for_overwrite<unsigned[]>(4 * kCapacity);
				unsigned *const out[4] = {scratch.get(), scratch.get() + kCapacity, scratch.get() + 2 * kCapacity, scratch.get() + 3 * kCapacity};
				const auto parsed = chunk_blocks(p, in.end, count - done, out, emit);
				if (!parsed) {
					chunks = false;
					break;
				}
				done += parsed;
			}
			unsigned tail[64];
			std::size_t used = 0;
			while (done + 2 <= count && in.end - p >= 32) {
				const char *next = p;
				auto invalid = _mm256_setzero_si256();
				const auto groups = two(next, invalid);
				if (!_mm256_testz_si256(invalid, invalid)) break;
				const auto halves = _mm256_madd_epi16(_mm256_packus_epi32(groups, groups), _mm256_set1_epi32(0x00012710));
				const unsigned h0 = _mm256_extract_epi32(halves, 0), l0 = _mm256_extract_epi32(halves, 1);
				const unsigned h1 = _mm256_extract_epi32(halves, 4), l1 = _mm256_extract_epi32(halves, 5);
				if (h0 > 42 || (h0 == 42 && l0 > 94967295) || h1 > 42 || (h1 == 42 && l1 > 94967295)) break;
				tail[used++] = h0 * 100000000 + l0;
				tail[used++] = h1 * 100000000 + l1;
				if (used == 64) {
					emit(tail, used);
					used = 0;
				}
				done += 2;
				p = next;
			}
			if (used) emit(tail, used);
			in.current = p;
			if (done < count) {
				unsigned value = 0;
				in.read(value);
				emit(&value, 1);
				++done;
			}
		}
	}

	static void read(FastInput &in, unsigned *out, std::size_t count) {
		read_blocks(in, count, [&](const unsigned *values, std::size_t n) {
			std::memcpy(out, values, n * sizeof(unsigned));
			out += n;
		});
	}

};
}
#pragma GCC pop_options
#endif


namespace fast_io_detail {
template <class Emit, std::size_t... I>
void emit_record(Emit &emit, const std::uint32_t *values, std::index_sequence<I...>) {
	emit(values[I]...);
}
}

template <class Emit>
void read_unsigned_blocks(FastInput &input, std::size_t count, Emit emit) {
#if defined(__linux__) && defined(__x86_64__) && defined(__GNUC__) && !defined(__clang__)
	if (__builtin_cpu_supports("avx2")) {
		fast_io_detail::BulkInput::read_blocks(input, count, emit);
		return;
	}
#endif
	std::uint32_t values[64];
	while (count) {
		const auto n = std::min<std::size_t>(count, 64);
		for (std::size_t i = 0; i < n; ++i) input.read(values[i]);
		emit(values, n);
		count -= n;
	}
}

inline void read_unsigned(FastInput &input, std::span<std::uint32_t> out) {
	read_unsigned_blocks(input, out.size(), [&](const std::uint32_t *values, std::size_t count) {
		std::memcpy(out.data(), values, count * sizeof(std::uint32_t));
		out = out.subspan(count);
	});
}

template <std::size_t Fields, class Emit>
void read_unsigned_records(FastInput &input, std::size_t count, Emit emit) {
	static_assert(Fields > 0);
	if (count > std::numeric_limits<std::size_t>::max() / Fields) throw std::length_error("input size");
	std::array<std::uint32_t, Fields> pending;
	std::size_t used = 0;
	constexpr auto indices = std::make_index_sequence<Fields>();
	read_unsigned_blocks(input, count * Fields, [&](const std::uint32_t *values, std::size_t n) {
		std::size_t i = 0;
		if (used) {
			while (used < Fields && i < n) pending[used++] = values[i++];
			if (used == Fields) {
				fast_io_detail::emit_record(emit, pending.data(), indices);
				used = 0;
			}
		}
		for (; i < n && n - i >= Fields; i += Fields) fast_io_detail::emit_record(emit, values + i, indices);
		while (i < n) pending[used++] = values[i++];
	});
}


#line 1 "components/fps_output.hpp"



#line 1 "components/mint_arithmetic.hpp"



#line 1 "math/mint.hpp"


#line 4 "math/mint.hpp"
#include <istream>
#include <ostream>
#line 7 "math/mint.hpp"

#if defined(__GNUC__) && !defined(__clang__) && (defined(__x86_64__) || defined(__i386__))
#pragma GCC push_options
#pragma GCC optimize("O3")
#endif

namespace mint_detail {
template <class I>
constexpr bool mint_integral = std::is_integral_v<I>
#if defined(__SIZEOF_INT128__)
							   || std::is_same_v<I, __int128> || std::is_same_v<I, unsigned __int128>
#endif
	;
struct [[gnu::warn_unused]] Mint {
	using mint = Mint;
	unsigned v = 0;
	[[nodiscard]] static constexpr unsigned mod() { return 998244353; }
	[[nodiscard, gnu::always_inline]] static unsigned reduce(unsigned long long x) {
		return (x + (unsigned long long)(unsigned(x) * 998244351u) * mod()) >> 32;
	}
	[[gnu::always_inline]] Mint() = default;
	template <class I, std::enable_if_t<mint_integral<I>, int> = 0>
	Mint(I x) {
		long long r = x % (long long)mod();
		if (r < 0) r += mod();
		v = reduce((unsigned long long)r * 932051910);
	}
	[[nodiscard, gnu::always_inline]] static Mint raw(unsigned x) {
		Mint a;
		a.v = reduce((unsigned long long)x * 932051910);
		return a;
	}
	[[nodiscard, gnu::always_inline]] static Mint word(unsigned x) {
		Mint a;
		a.v = x;
		return a;
	}
	[[nodiscard, gnu::always_inline]] unsigned val() const {
		unsigned r = reduce(v);
		return r < mod() ? r : r - mod();
	}
	friend std::istream &operator>>(std::istream &is, Mint &x) {
		long long value;
		if (is >> value) x = Mint(value);
		return is;
	}
	friend std::ostream &operator<<(std::ostream &os, const Mint &x) {
		return os << x.val();
	}
	[[gnu::always_inline]] Mint &operator+=(Mint b) {
		v += b.v;
		if (v >= 2 * mod()) v -= 2 * mod();
		return *this;
	}
	[[gnu::always_inline]] Mint &operator-=(Mint b) {
		v -= b.v;
		if (v >> 31) v += 2 * mod();
		return *this;
	}
	[[gnu::always_inline]] Mint &operator*=(Mint b) {
		v = reduce((unsigned long long)v * b.v);
		return *this;
	}
	Mint &operator/=(Mint b) { return *this *= b.inv(); }
	Mint &operator++() { return *this += Mint(1); }
	Mint &operator--() { return *this -= Mint(1); }
	Mint operator++(int) {
		Mint a = *this;
		++*this;
		return a;
	}
	Mint operator--(int) {
		Mint a = *this;
		--*this;
		return a;
	}
	[[nodiscard, gnu::always_inline]] Mint operator+() const { return *this; }
	[[nodiscard, gnu::always_inline]] Mint operator-() const { return word(v ? 2 * mod() - v : 0); }
	[[nodiscard, gnu::always_inline]] Mint operator+(Mint b) const { return Mint(*this) += b; }
	[[nodiscard, gnu::always_inline]] Mint operator-(Mint b) const { return Mint(*this) -= b; }
	[[nodiscard, gnu::always_inline]] Mint operator*(Mint b) const { return Mint(*this) *= b; }
	[[nodiscard]] Mint operator/(Mint b) const { return Mint(*this) /= b; }
	[[nodiscard, gnu::always_inline]] bool operator==(Mint b) const { return (v < mod() ? v : v - mod()) == (b.v < mod() ? b.v : b.v - mod()); }
	[[nodiscard, gnu::always_inline]] bool operator!=(Mint b) const { return !(*this == b); }
	template <class I, std::enable_if_t<mint_integral<I>, int> = 0>
	[[nodiscard]] friend Mint operator+(I a, Mint b) { return Mint(a) + b; }
	template <class I, std::enable_if_t<mint_integral<I>, int> = 0>
	[[nodiscard]] friend Mint operator-(I a, Mint b) { return Mint(a) - b; }
	template <class I, std::enable_if_t<mint_integral<I>, int> = 0>
	[[nodiscard]] friend Mint operator*(I a, Mint b) { return Mint(a) * b; }
	template <class I, std::enable_if_t<mint_integral<I>, int> = 0>
	[[nodiscard]] friend Mint operator/(I a, Mint b) { return Mint(a) / b; }
	template <class I, std::enable_if_t<mint_integral<I>, int> = 0>
	[[nodiscard]] friend bool operator==(I a, Mint b) { return Mint(a) == b; }
	template <class I, std::enable_if_t<mint_integral<I>, int> = 0>
	[[nodiscard]] friend bool operator!=(I a, Mint b) { return Mint(a) != b; }
	[[nodiscard]] Mint pow(long long n) const {
		assert(n >= 0);
		if (!n) return Mint(1);
		Mint r(1), a = *this;
		for (; n > 1; n >>= 1, a *= a)
			if (n & 1) r *= a;
		return r * a;
	}
	[[nodiscard, gnu::noinline]] Mint inv() const {
		assert(*this != Mint(0));
		Mint a2 = *this * *this, a3 = a2 * *this;
		Mint a7 = a3 * a3 * *this;
		Mint a31 = a7;
		for (int i = 0; i < 2; ++i) a31 *= a31;
		a31 *= a3;
		Mint r = a7;
		for (int i = 0; i < 3; ++i) r *= r;
		r *= a3;
		r *= r;
		for (int j = 0; j < 4; ++j) {
			for (int i = 0; i < 5; ++i) r *= r;
			r *= a31;
		}
		for (int i = 0; i < 3; ++i) r *= r;
		return r * a7;
	}
};
} // namespace mint_detail

using mint = mint_detail::Mint;
using fps_mint = mint;

#if defined(__GNUC__) && !defined(__clang__) && (defined(__x86_64__) || defined(__i386__))
#pragma GCC pop_options
#endif


#line 5 "components/mint_arithmetic.hpp"
#if defined(__x86_64__) || defined(__i386__)
#include <immintrin.h>
#endif

#if defined(__GNUC__) && !defined(__clang__) && (defined(__x86_64__) || defined(__i386__))
#pragma GCC push_options
#pragma GCC optimize("O3")
#endif

namespace mint_detail {
struct Ops {
	using T = mint;
	typedef unsigned int U __attribute__((may_alias));
	static constexpr U mod = 998244353, twice = 2 * mod, one = 301989884;
	static U reduce(unsigned long long x) { return T::reduce(x); }
	static U mul(U a, U b) { return reduce((unsigned long long)a * b); }
	static U add(U a, U b) {
		U c = a + b;
		return c < twice ? c : c - twice;
	}
	static U sub(U a, U b) {
		U c = a - b;
		return c < twice ? c : c + twice;
	}
	static U in(U a) { return mul(a, 932051910); }
	static U out(U a) {
		a = reduce(a);
		return a < mod ? a : a - mod;
	}
	static U quotient(U r) { return ((unsigned long long)r << 32) / mod; }
	static U fixed_mul(U a, U r, U q) {
		return a * r - U(((unsigned long long)a * q) >> 32) * mod;
	}

#if defined(__x86_64__) || defined(__i386__)
	static bool vectorized() {
#if defined(__AVX2__)
		return true;
#else
		static const bool available = __builtin_cpu_supports("avx2");
		return available;
#endif
	}
	using I8 = __m256i;
	__attribute__((target("avx2"))) static I8 load8(const U *a) {
		return _mm256_loadu_si256(reinterpret_cast<const I8 *>(a));
	}
	__attribute__((target("avx2"))) static void store8(U *a, I8 x) {
		_mm256_storeu_si256(reinterpret_cast<I8 *>(a), x);
	}
	__attribute__((target("avx2"), always_inline)) static inline I8 reduce8(I8 x, I8 y) {
		I8 r = _mm256_set1_epi32(998244351), p = _mm256_set1_epi32(mod);
		x = _mm256_add_epi64(x, _mm256_mul_epu32(_mm256_mul_epu32(x, r), p));
		y = _mm256_add_epi64(y, _mm256_mul_epu32(_mm256_mul_epu32(y, r), p));
		return _mm256_blend_epi32(_mm256_srli_epi64(x, 32), y, 0xaa);
	}
	__attribute__((target("avx2"))) static I8 mul8(I8 a, I8 b) {
		return reduce8(_mm256_mul_epu32(a, b), _mm256_mul_epu32(_mm256_srli_epi64(a, 32), _mm256_srli_epi64(b, 32)));
	}
	__attribute__((target("avx2"))) static I8 fixed_mul8(I8 a, I8 r, I8 q) {
		I8 x = _mm256_srli_epi64(_mm256_mul_epu32(a, q), 32);
		I8 y = _mm256_mul_epu32(_mm256_srli_epi64(a, 32), q);
		I8 t = _mm256_blend_epi32(x, y, 0xaa);
		return _mm256_sub_epi32(_mm256_mullo_epi32(a, r),
								_mm256_mullo_epi32(t, _mm256_set1_epi32(mod)));
	}
	__attribute__((target("avx2"))) static I8 fixed_even8(I8 a, I8 r, I8 q) {
		I8 t = _mm256_srli_epi64(_mm256_mul_epu32(a, q), 32);
		return _mm256_sub_epi64(_mm256_mul_epu32(a, r),
								_mm256_mul_epu32(t, _mm256_set1_epi32(mod)));
	}
	__attribute__((target("avx2"))) static I8 add8(I8 a, I8 b) {
		I8 x = _mm256_add_epi32(a, b);
		return _mm256_min_epu32(x, _mm256_sub_epi32(x, _mm256_set1_epi32(twice)));
	}
	__attribute__((target("avx2"))) static I8 sub8(I8 a, I8 b) {
		I8 x = _mm256_sub_epi32(a, b);
		return _mm256_min_epu32(x, _mm256_add_epi32(x, _mm256_set1_epi32(twice)));
	}

#endif
};
}

#if defined(__GNUC__) && !defined(__clang__) && (defined(__x86_64__) || defined(__i386__))
#pragma GCC pop_options
#endif


#line 10 "components/fps_output.hpp"
#include <vector>

#if defined(__linux__) && defined(__x86_64__) && defined(__GNUC__) && !defined(__clang__)
#include <immintrin.h>
#include <sys/mman.h>
#pragma GCC push_options
_Pragma("GCC target(\"avx2\")")
#pragma GCC optimize("O3")

namespace fps_detail {

inline void prepare_fps_output(std::vector<mint> &out, std::size_t n) {
	if (n <= out.capacity()) return;
	out.reserve(n);
	if (n - out.size() < (1u << 17)) return;
	const auto first = reinterpret_cast<std::uintptr_t>(out.data() + out.size());
	const auto end = reinterpret_cast<std::uintptr_t>(out.data() + n);
	[[maybe_unused]] const auto advise = [first, end](std::uintptr_t alignment, int advice) {
		const auto lo = (first + alignment - 1) & ~(alignment - 1), hi = end & ~(alignment - 1);
		if (lo < hi) static_cast<void>(madvise(reinterpret_cast<void *>(lo), hi - lo, advice));
	};
#if defined(MADV_HUGEPAGE)
	advise(2u * 1024 * 1024, MADV_HUGEPAGE);
#endif
#if defined(MADV_POPULATE_WRITE)
	advise(4096, MADV_POPULATE_WRITE);
#endif
}

template <bool montgomery>
inline void append_fps_words(std::vector<mint> &out, const unsigned *in, std::size_t n) {
	static_assert(sizeof(mint) == sizeof(unsigned) && std::is_trivially_copyable_v<mint>);
	if (n == 0) return;
	if (n > out.max_size() - out.size()) throw std::length_error("fps output");
	prepare_fps_output(out, out.size() + n);
	std::array<mint, 64> batch;
	std::size_t i = 0;
	for (; i + 64 <= n; i += 64) {
		if constexpr (montgomery) std::memcpy(static_cast<void *>(batch.data()), in + i, sizeof(batch));
		else {
#pragma GCC unroll 8
			for (int j = 0; j < 64; j += 8) {
				const auto x = mint_detail::Ops::mul8(_mm256_loadu_si256(reinterpret_cast<const __m256i *>(in + i + j)), _mm256_set1_epi32(932051910u));
				std::memcpy(static_cast<void *>(batch.data() + j), &x, sizeof(x));
			}
		}
		out.insert(out.end(), batch.begin(), batch.end());
	}
	for (; i < n; ++i) {
		if constexpr (montgomery) out.emplace_back(mint::word(in[i]));
		else out.emplace_back(mint::raw(in[i]));
	}
}
}
#pragma GCC pop_options
#endif

#line 1 "components/fps_write.hpp"



#line 10 "components/fps_write.hpp"

#if defined(__linux__) && defined(__x86_64__) && defined(__GNUC__) && !defined(__clang__)
#include <immintrin.h>
#pragma GCC push_options
_Pragma("GCC target(\"avx2\")")
#pragma GCC optimize("O3")
namespace fps_io_fields {

namespace detail {

struct alignas(32) Lanes {
	std::uint32_t lane[8];
};

constexpr Lanes splat(std::uint32_t x) { return {{x, x, x, x, x, x, x, x}}; }
constexpr Lanes all16(std::uint32_t x) { return splat(x * 0x10001); }

constexpr int Z = 0x80;

constexpr Lanes shuffle(const int (&b)[16]) {
	Lanes t{};
	for (int i = 0; i < 32; ++i) t.lane[i / 4] |= std::uint32_t(b[i % 16] & 0xFF) << 8 * (i % 4);
	return t;
}

struct Constants {
	Lanes div10 = splat(429496730);
	Lanes div10e5 = splat(1407374884);
	Lanes pack = splat(1 - 10000 * 65536u);
	Lanes ten = splat(10);
	Lanes tail = splat(' ' << 8 | '0');
	Lanes by100 = all16(5243), hundred = all16(100);
	Lanes by10 = all16(6554);
	Lanes tens_out = all16(2559);
	Lanes ones = splat(~0u), zero = splat(0x30303030), space = splat(0x20202020);

	Lanes chunk0[2] = {shuffle({0, 1, 2, 3, 4, 5, 6, 7, Z, Z, 8, 9, 10, 11, 12, 13}),
					shuffle({Z, Z, Z, Z, Z, Z, Z, Z, 0, 1, Z, Z, Z, Z, Z, Z})};
	Lanes chunk1[3] = {shuffle({14, 15, Z, Z, Z, Z, Z, Z, Z, Z, Z, Z, Z, Z, Z, Z}),
					shuffle({Z, Z, 4, 5, Z, Z, Z, Z, Z, Z, Z, Z, 8, 9, Z, Z}),
					shuffle({Z, Z, Z, Z, 0, 1, 2, 3, 4, 5, 6, 7, Z, Z, 8, 9})};
	Lanes chunk2[3] = {shuffle({10, 11, 12, 13, 14, 15, Z, Z, Z, Z, Z, Z, Z, Z, Z, Z}),
					shuffle({Z, Z, Z, Z, Z, Z, 12, 13, Z, Z, Z, Z, Z, Z, Z, Z}),
					shuffle({Z, Z, Z, Z, Z, Z, Z, Z, 0, 1, 2, 3, 4, 5, 6, 7})};
	Lanes chunk3[3] = {shuffle({0, 1, Z, Z, Z, Z, Z, Z, Z, Z, 4, 5, Z, Z, Z, Z}),
					shuffle({Z, Z, 8, 9, 10, 11, 12, 13, 14, 15, Z, Z, Z, Z, Z, Z}),
					shuffle({Z, Z, Z, Z, Z, Z, Z, Z, Z, Z, Z, Z, 0, 1, 2, 3})};
	Lanes chunk4[2] = {shuffle({4, 5, 6, 7, Z, Z, 8, 9, 10, 11, 12, 13, 14, 15, Z, Z}),
					shuffle({Z, Z, Z, Z, 8, 9, Z, Z, Z, Z, Z, Z, Z, Z, 12, 13})};
};

inline constexpr Constants kConstants{};

[[gnu::always_inline]] inline __m256i load(const Lanes& l) {
	return _mm256_load_si256(reinterpret_cast<const __m256i*>(&l));
}

[[gnu::always_inline]] inline __m256i shuffle(__m256i x, const Lanes& control) {
	return _mm256_shuffle_epi8(x, load(control));
}

struct Divided {
	__m256i halves, tail;
};

[[gnu::always_inline]] inline Divided divide(__m256i v, __m256i odd, const Constants& k) {
	const __m256i w = _mm256_blend_epi32(_mm256_srli_epi64(_mm256_mul_epu32(v, load(k.div10)), 32),
										_mm256_mul_epu32(odd, load(k.div10)), 0xAA);
	const __m256i h = _mm256_blend_epi32(_mm256_srli_epi64(_mm256_mul_epu32(v, load(k.div10e5)), 47),
										_mm256_srli_epi64(_mm256_mul_epu32(odd, load(k.div10e5)), 15), 0xAA);
	const __m256i halves = _mm256_add_epi32(_mm256_slli_epi32(w, 16), _mm256_mullo_epi32(h, load(k.pack)));
	const __m256i tail = _mm256_sub_epi32(_mm256_add_epi32(v, load(k.tail)), _mm256_mullo_epi32(w, load(k.ten)));
	return {halves, tail};
}

[[gnu::always_inline]] inline __m256i two_digits(__m256i z, const Constants& k) {
	const __m256i tens = _mm256_mulhi_epu16(z, load(k.by10));
	return _mm256_sub_epi16(_mm256_slli_epi16(z, 8), _mm256_mullo_epi16(tens, load(k.tens_out)));
}

[[gnu::always_inline]] inline __m256i text(__m256i digits, const Constants& k) {
	const __m256i upto = _mm256_xor_si256(digits, _mm256_add_epi64(digits, load(k.ones)));
	return _mm256_blendv_epi8(_mm256_or_si256(digits, load(k.zero)), load(k.space), upto);
}

[[gnu::always_inline]] inline void digits(const Divided& d, __m256i& lo, __m256i& hi, const Constants& k) {
	const __m256i hundreds = _mm256_srli_epi16(_mm256_mulhi_epu16(d.halves, load(k.by100)), 3);
	const __m256i rest = _mm256_sub_epi16(d.halves, _mm256_mullo_epi16(hundreds, load(k.hundred)));
	lo = text(two_digits(_mm256_unpacklo_epi16(hundreds, rest), k), k);
	hi = text(two_digits(_mm256_unpackhi_epi16(hundreds, rest), k), k);
}

[[gnu::always_inline]] inline void store(char* p, const Divided& lo, const Divided& hi, const Constants& k) {
	__m256i d01, d23, d45, d67;
	const auto either = [&](__m256i a, __m256i b) __attribute__((target("avx2"))) { return _mm256_or_si256(a, b); };
	digits(lo, d01, d23, k);
	digits(hi, d45, d67, k);
	const __m256i c0 = either(shuffle(d01, k.chunk0[0]), shuffle(lo.tail, k.chunk0[1]));
	const __m256i c1 = either(either(shuffle(d01, k.chunk1[0]), shuffle(lo.tail, k.chunk1[1])), shuffle(d23, k.chunk1[2]));
	const __m256i c2 = either(either(shuffle(d23, k.chunk2[0]), shuffle(lo.tail, k.chunk2[1])), shuffle(d45, k.chunk2[2]));
	const __m256i c3 = either(either(shuffle(hi.tail, k.chunk3[0]), shuffle(d45, k.chunk3[1])), shuffle(d67, k.chunk3[2]));
	const __m256i c4 = either(shuffle(d67, k.chunk4[0]), shuffle(hi.tail, k.chunk4[1]));

	const auto whole = [p](int c, __m256i chunk) __attribute__((target("avx2"))) { _mm256_storeu_si256(reinterpret_cast<__m256i*>(p + 64 + 16 * c), chunk); };
	const auto low = [p](int c, __m256i chunk) __attribute__((target("avx2"))) { _mm_store_si128(reinterpret_cast<__m128i*>(p + 16 * c), _mm256_castsi256_si128(chunk)); };
	whole(4, c4), whole(3, c3), whole(2, c2), whole(1, c1), whole(0, c0);
	low(0, c0), low(1, c1), low(2, c2), low(3, c3), low(4, c4);
}

[[gnu::always_inline]] inline void divide16(const std::uint32_t* x, Divided& lo, Divided& hi, const Constants& k) {
	const auto at = [x](int i) __attribute__((target("avx2"))) { return _mm256_loadu_si256(reinterpret_cast<const __m256i*>(x + i)); };
	const auto both = [x](int i) __attribute__((target("avx2"))) { return _mm256_broadcastsi128_si256(_mm_loadu_si128(reinterpret_cast<const __m128i*>(x + i))); };
	const __m256i l = _mm256_blend_epi32(at(0), both(8), 0xF0), h = _mm256_blend_epi32(at(8), both(4), 0x0F);
	lo = divide(l, _mm256_srli_epi64(l, 32), k);
	hi = divide(h, _mm256_srli_epi64(h, 32), k);
}

inline void format(char* p, const std::uint32_t* x, std::size_t count, const Constants& k) {
	Divided dl, dh;
	divide16(x, dl, dh, k);
	for (std::size_t i = 16; i < count; i += 16, p += 160) {
		Divided nl, nh;
		divide16(x + i, nl, nh, k);
		store(p, dl, dh, k);
		dl = nl, dh = nh;
	}
	store(p, dl, dh, k);
}

}

inline constexpr std::size_t kBlock = 25600;
inline constexpr std::size_t kTextBytes = 10 * kBlock;

}

namespace fps_detail {
[[gnu::always_inline]] inline __m256i io_values(__m256i x) {
	const auto p = _mm256_set1_epi32(mint::mod()), ni = _mm256_set1_epi32(998244351);
	auto even = _mm256_and_si256(x, _mm256_set1_epi64x(0xffffffffULL));
	auto odd = _mm256_srli_epi64(x, 32);
	even = _mm256_add_epi64(even, _mm256_mul_epu32(_mm256_mul_epu32(even, ni), p));
	odd = _mm256_add_epi64(odd, _mm256_mul_epu32(_mm256_mul_epu32(odd, ni), p));
	const auto value = _mm256_blend_epi32(_mm256_srli_epi64(even, 32), odd, 0xaa);
	return _mm256_min_epu32(value, _mm256_sub_epi32(value, p));
}

inline void write_coefficients_native(FastOutput &output, const mint *in, std::size_t n) {
	alignas(32) unsigned values[25600];
	alignas(32) char text[256000];
	const auto &k = fps_io_fields::detail::kConstants;
	for (std::size_t i = 0; i < n; i += 25600) {
		const auto count = std::min<std::size_t>(25600, n - i), full = count / 16 * 16;
		char *end = text;
		static constexpr char zeros[] = "0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ";
		const auto empty = [&]() __attribute__((target("avx2"))) {
			std::size_t at = 0;
			const auto p = _mm256_set1_epi32(mint::mod());
			for (; at + 8 <= count; at += 8) {
				__m256i x;
				std::memcpy(&x, in + i + at, sizeof(x));
				x = _mm256_min_epu32(x, _mm256_sub_epi32(x, p));
				if (!_mm256_testz_si256(x, x)) return false;
			}
			for (; at < count; ++at) if (in[i + at] != mint(0)) return false;
			return true;
		};
		if (empty()) {
			for (std::size_t at = 0; at < count; at += 16) std::memcpy(text + 2 * at, zeros, 32);
			end += 2 * count;
		} else {
			std::size_t j = 0;
			for (; j + 8 <= count; j += 8) {
				alignas(32) unsigned words[8];
				std::memcpy(words, in + i + j, sizeof(words));
				_mm256_storeu_si256(reinterpret_cast<__m256i *>(values + j), io_values(_mm256_load_si256(reinterpret_cast<const __m256i *>(words))));
			}
			for (; j < count; ++j) values[j] = in[i + j].val();
			auto zero = [&](std::size_t at) __attribute__((target("avx2"))) {
				auto x = _mm256_or_si256(_mm256_loadu_si256(reinterpret_cast<const __m256i *>(values + at)), _mm256_loadu_si256(reinterpret_cast<const __m256i *>(values + at + 8)));
				return _mm256_testz_si256(x, x);
			};
			std::size_t at = 0;
			while (at < full) {
				if (zero(at)) { std::memcpy(end, zeros, 32); end += 32; at += 16; }
				else {
					const auto first = at;
					do at += 16; while (at < full && !zero(at));
					fps_io_fields::detail::format(end, values + first, at - first, k);
					end += 10 * (at - first);
				}
			}
			if (full < count) {
				alignas(32) unsigned rest[16] = {};
				alignas(16) char last[160];
				std::copy_n(values + full, count - full, rest);
				bool empty = true;
				for (auto x : rest) empty &= x == 0;
				if (empty) { std::memcpy(end, zeros, 2 * (count - full)); end += 2 * (count - full); }
				else {
					fps_io_fields::detail::format(last, rest, 16, k);
					std::memcpy(end, last, 10 * (count - full));
					end += 10 * (count - full);
				}
			}
		}
		if (i + count == n) end[-1] = '\n';
		fast_io_detail::BulkOutput::write(output, std::string_view(text, end - text));
	}
	if (!n) output.write('\n');
}
}
#pragma GCC pop_options
#endif

namespace fps_detail {
inline void write_coefficients(FastOutput &output, const mint *in, std::size_t n) {
#if defined(__linux__) && defined(__x86_64__) && defined(__GNUC__) && !defined(__clang__)
	if (__builtin_cpu_supports("avx2")) return write_coefficients_native(output, in, n);
#endif
	for (std::size_t i = 0; i < n; ++i) { output.write(in[i].val()); output.write(' '); }
	output.write('\n');
}
}

#line 8 "components/fps_io.hpp"

#if defined(__linux__) && defined(__x86_64__) && defined(__GNUC__) && !defined(__clang__)
#include <immintrin.h>
#pragma GCC push_options
_Pragma("GCC target(\"avx2\")")
#pragma GCC optimize("O3")

namespace fps_detail {
inline void read_coefficients(FastInput &input, std::vector<mint> &out, std::size_t n) {
	if (!n) return;
	if (n > out.max_size() - out.size()) throw std::length_error("fps input");
	prepare_fps_output(out, out.size() + n);
	fast_io_detail::BulkInput::read_blocks(input, n, [&](const unsigned *values, std::size_t count) {
		append_fps_words<false>(out, values, count);
	});
}
}
#pragma GCC pop_options
#endif