test/fps_light_sqrt.test.cpp

test/fps_light_sqrt.test.cpp · GitHub で見る

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#define PROBLEM "https://judge.yosupo.jp/problem/sqrt_of_formal_power_series"
#include "fixtures/fps_light_io.hpp"

int main() {
	FastInput input;
	FastOutput output;
	int n = 0; input.read(n);
	auto root = read_light_fps(input, n).sqrt();
	if (root.empty()) output.println(-1);
	else write_light_fps(output, root);
}

#line 1 "test/fps_light_sqrt.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/sqrt_of_formal_power_series"
#line 1 "test/fixtures/fps_light_io.hpp"



#line 1 "fps/fps_light.hpp"


#line 1 "components/fps_light_backend.hpp"



#line 1 "components/fps_light_core.hpp"



#include <algorithm>
#include <bit>
#include <cassert>
#include <cstddef>
#include <initializer_list>
#include <limits>
#include <optional>
#include <type_traits>
#include <utility>
#include <vector>
#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 15 "components/fps_light_core.hpp"

#line 1 "components/fps_light_ntt.hpp"



#line 11 "components/fps_light_ntt.hpp"

#if (defined(__x86_64__) || defined(__i386__)) && (defined(__GNUC__) || defined(__clang__))
#include <immintrin.h>
#endif

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

namespace fps_light_detail {
using std::size_t;
using Series = std::vector<mint>;
static_assert(sizeof(mint) == sizeof(unsigned) && std::is_trivially_copyable_v<mint>);
inline bool simd_available() {
#if (defined(__x86_64__) || defined(__i386__)) && (defined(__GNUC__) || defined(__clang__))
	return __builtin_cpu_supports("avx2");
#else
	return false;
#endif
}
template<bool Inverse>
void scalar_stage(mint *a, size_t n, size_t h, const mint *roots) {
	for (size_t start = 0; start < n; start += h * 2)
		for (size_t j = 0; j < h; ++j) {
			mint x = a[start + j], y = a[start + j + h];
			if constexpr (Inverse) y *= roots[j];
			a[start + j] = x + y;
			a[start + j + h] = x - y;
			if constexpr (!Inverse) a[start + j + h] *= roots[j];
		}
}

#if (defined(__x86_64__) || defined(__i386__)) && (defined(__GNUC__) || defined(__clang__))
__attribute__((target("avx2")))
inline __m256i product(__m256i x, __m256i y) {
	const auto p = _mm256_set1_epi32(mint::mod());
	auto even = _mm256_mul_epu32(x, y);
	auto odd = _mm256_mul_epu32(_mm256_srli_epi64(x, 32), _mm256_srli_epi64(y, 32));
	auto q = _mm256_mullo_epi32(_mm256_mullo_epi32(x, y), _mm256_set1_epi32(998244351));
	even = _mm256_srli_epi64(_mm256_add_epi64(even, _mm256_mul_epu32(q, p)), 32);
	odd = _mm256_add_epi64(odd, _mm256_mul_epu32(_mm256_srli_epi64(q, 32), p));
	return _mm256_blend_epi32(even, odd, 0xaa);
}

__attribute__((target("avx2")))
inline __m256i fixed_product(__m256i x, __m256i w, __m256i q) {
	auto even = _mm256_mul_epu32(x, q);
	auto odd = _mm256_mul_epu32(_mm256_srli_epi64(x, 32), _mm256_srli_epi64(q, 32));
	q = _mm256_blend_epi32(_mm256_srli_epi64(even, 32), odd, 0xaa);
	return _mm256_sub_epi32(_mm256_mullo_epi32(x, w), _mm256_mullo_epi32(q, _mm256_set1_epi32(mint::mod())));
}

__attribute__((target("avx2")))
inline void pointwise_avx2(mint *a, const mint *b, size_t n) {
	size_t i = 0;
	for (; i + 8 <= n; i += 8) {
		auto x = _mm256_loadu_si256(reinterpret_cast<const __m256i *>(a + i));
		auto y = _mm256_loadu_si256(reinterpret_cast<const __m256i *>(b + i));
		_mm256_storeu_si256(reinterpret_cast<__m256i *>(a + i), product(x, y));
	}
	for (; i < n; ++i) a[i] *= b[i];
}

__attribute__((target("avx2")))
inline void scale_avx2(mint *a, size_t n, mint factor) {
	size_t i = 0;
	unsigned value = factor.val();
	const auto f = _mm256_set1_epi32(value);
	const auto q = _mm256_set1_epi32((static_cast<unsigned long long>(value) << 32) / mint::mod());
	for (; i + 8 <= n; i += 8) {
		auto x = _mm256_loadu_si256(reinterpret_cast<const __m256i *>(a + i));
		_mm256_storeu_si256(reinterpret_cast<__m256i *>(a + i), fixed_product(x, f, q));
	}
	for (; i < n; ++i) a[i] *= factor;
}

__attribute__((target("avx2")))
inline __m256i sum_vec(__m256i x, __m256i y) {
	auto z = _mm256_add_epi32(x, y);
	return _mm256_min_epu32(z, _mm256_sub_epi32(z, _mm256_set1_epi32(2 * mint::mod())));
}

__attribute__((target("avx2")))
inline __m256i diff_vec(__m256i x, __m256i y) {
	auto z = _mm256_sub_epi32(x, y);
	return _mm256_min_epu32(z, _mm256_add_epi32(z, _mm256_set1_epi32(2 * mint::mod())));
}

template<int Half, bool Inverse>
__attribute__((target("avx2"), always_inline))
inline __m256i leaf_stage(__m256i x, const unsigned *w, const unsigned *q) {
	constexpr int mask = Half == 4 ? 0xf0 : Half == 2 ? 0xcc : 0xaa;
	__m256i factor{}, quotient{};
	if constexpr (Half == 4) {
		factor = _mm256_broadcastsi128_si256(_mm_loadu_si128(reinterpret_cast<const __m128i *>(w + 4)));
		quotient = _mm256_broadcastsi128_si256(_mm_loadu_si128(reinterpret_cast<const __m128i *>(q + 4)));
	}
	if constexpr (Half == 2) {
		factor = _mm256_broadcastq_epi64(_mm_loadl_epi64(reinterpret_cast<const __m128i *>(w + 2)));
		quotient = _mm256_broadcastq_epi64(_mm_loadl_epi64(reinterpret_cast<const __m128i *>(q + 2)));
	}
	if constexpr (Inverse && Half > 1) x = _mm256_blend_epi32(x, fixed_product(x, factor, quotient), mask);
	__m256i y;
	if constexpr (Half == 4) y = _mm256_permute2x128_si256(x, x, 1);
	if constexpr (Half == 2) y = _mm256_shuffle_epi32(x, 0x4e);
	if constexpr (Half == 1) y = _mm256_shuffle_epi32(x, 0xb1);
	auto high = diff_vec(y, x);
	if constexpr (!Inverse && Half > 1) high = fixed_product(high, factor, quotient);
	return _mm256_blend_epi32(sum_vec(x, y), high, mask);
}

template<bool Inverse>
__attribute__((target("avx2")))
void leaf8(mint *a, size_t n, const mint *roots) {
	alignas(32) unsigned values[8], quotients[8];
	for (int i = 0; i < 8; ++i) {
		values[i] = roots[i].val();
		quotients[i] = (static_cast<unsigned long long>(values[i]) << 32) / mint::mod();
	}
	for (size_t i = 0; i < n; i += 8) {
		auto x = _mm256_loadu_si256(reinterpret_cast<const __m256i *>(a + i));
		x = leaf_stage<Inverse ? 1 : 4, Inverse>(x, values, quotients);
		x = leaf_stage<2, Inverse>(x, values, quotients);
		x = leaf_stage<Inverse ? 4 : 1, Inverse>(x, values, quotients);
		_mm256_storeu_si256(reinterpret_cast<__m256i *>(a + i), x);
	}
}

__attribute__((target("avx2")))
inline void expand_roots(mint *out, const mint *previous, size_t n, mint step) {
	for (size_t i = 0; i < n; ++i) {
		out[2 * i].v = previous[i].v;
		out[2 * i + 1].v = mint::reduce(static_cast<unsigned long long>(previous[i].v) * step.v);
	}
}

template<bool Inverse>
__attribute__((target("avx2")))
void vector_stage(mint *a, size_t n, size_t h, const mint *roots) {
	for (size_t start = 0; start < n; start += h * 2)
		for (size_t j = 0; j < h; j += 8) {
			auto *first = reinterpret_cast<__m256i *>(a + start + j);
			auto *last = reinterpret_cast<__m256i *>(a + start + j + h);
			auto x = _mm256_loadu_si256(first), y = _mm256_loadu_si256(last);
			auto w = _mm256_loadu_si256(reinterpret_cast<const __m256i *>(roots + j));
			if constexpr (Inverse) y = product(y, w);
			auto sum = sum_vec(x, y), difference = diff_vec(x, y);
			if constexpr (!Inverse) difference = product(difference, w);
			_mm256_storeu_si256(first, sum);
			_mm256_storeu_si256(last, difference);
		}
}
#endif

struct Roots {
	Series forward{0, 1}, inverse{0, 1};
	void prepare(size_t n) {
		size_t begin = forward.size();
		if (begin >= n) return;
		forward.resize(n), inverse.resize(n);
		for (size_t h = begin; h < n; h *= 2) {
			mint step = mint(3).pow((mint::mod() - 1) / (h * 2));
#if (defined(__x86_64__) || defined(__i386__)) && (defined(__GNUC__) || defined(__clang__))
			if (simd_available() && h >= 16) expand_roots(forward.data() + h, forward.data() + h / 2, h / 2, step);
			else
#endif
			for (size_t j = 0; j < h / 2; ++j) {
				forward[h + 2 * j] = forward[h / 2 + j];
				forward[h + 2 * j + 1] = forward[h / 2 + j] * step;
			}
			inverse[h] = 1;
			for (size_t j = 1; j < h; ++j) inverse[h + j] = -forward[2 * h - j];
		}
	}
};

inline void pointwise(Series &a, const Series &b, [[maybe_unused]] bool vectorized = simd_available()) {
	assert(a.size() <= b.size());
#if (defined(__x86_64__) || defined(__i386__)) && (defined(__GNUC__) || defined(__clang__))
	if (vectorized) return pointwise_avx2(a.data(), b.data(), a.size());
#endif
	for (size_t i = 0; i < a.size(); ++i) a[i] *= b[i];
}

inline void scale(Series &a, mint factor, [[maybe_unused]] bool vectorized = simd_available()) {
#if (defined(__x86_64__) || defined(__i386__)) && (defined(__GNUC__) || defined(__clang__))
	if (vectorized) return scale_avx2(a.data(), a.size(), factor);
#endif
	for (auto &x : a) x *= factor;
}

template<bool Inverse>
void stage(mint *a, size_t n, size_t h, const mint *roots, [[maybe_unused]] bool vectorized) {
#if (defined(__x86_64__) || defined(__i386__)) && (defined(__GNUC__) || defined(__clang__))
	if (vectorized && h >= 8) return vector_stage<Inverse>(a, n, h, roots);
#endif
	scalar_stage<Inverse>(a, n, h, roots);
}

inline void transform(Series &a, bool inverse = false) {
	size_t n = a.size();
	assert(std::has_single_bit(n) && n <= (size_t(1) << 23));
	static Roots roots;
	roots.prepare(n);
	bool vectorized = simd_available();
	if (inverse) {
		size_t h = 1;
#if (defined(__x86_64__) || defined(__i386__)) && (defined(__GNUC__) || defined(__clang__))
		if (vectorized && n >= 8) leaf8<true>(a.data(), n, roots.inverse.data()), h = 8;
#endif
		for (; h < n; h *= 2) stage<true>(a.data(), n, h, roots.inverse.data() + h, vectorized);
		scale(a, mint::raw(mint::mod() - (mint::mod() - 1) / n), vectorized);
	} else {
		size_t h = n / 2;
		for (; h >= 8; h /= 2) stage<false>(a.data(), n, h, roots.forward.data() + h, vectorized);
#if (defined(__x86_64__) || defined(__i386__)) && (defined(__GNUC__) || defined(__clang__))
		if (vectorized && n >= 8) return leaf8<false>(a.data(), n, roots.forward.data());
#endif
		for (; h; h /= 2) stage<false>(a.data(), n, h, roots.forward.data() + h, vectorized);
	}
}

inline Series multiply(const Series &a, const Series &b) {
	if (a.empty() || b.empty()) return {};
	[[maybe_unused]] constexpr size_t limit = size_t(1) << 23;
	assert(a.size() <= limit && b.size() <= limit && a.size() + b.size() - 1 <= limit);
	size_t length = a.size() + b.size() - 1;
	if (std::min(a.size(), b.size()) <= 32) {
		Series result(length);
		for (size_t i = 0; i < a.size(); ++i)
			for (size_t j = 0; j < b.size(); ++j) result[i + j] += a[i] * b[j];
		return result;
	}
	size_t n = std::bit_ceil(length);
	Series x = a;
	x.resize(n);
	transform(x);
	if (&a == &b) {
		pointwise(x, x);
	} else {
		Series y = b;
		y.resize(n);
		transform(y);
		pointwise(x, y);
	}
	transform(x, true);
	x.resize(length);
	return x;
}
}

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


#line 1 "components/fps_sparse_scalar.hpp"



#line 1 "components/fps_sparse_terms.hpp"



#line 7 "components/fps_sparse_terms.hpp"
#include <numeric>
#line 10 "components/fps_sparse_terms.hpp"

namespace fps_detail {
struct SparseTerms {
	using Series = std::vector<mint>;
	using Terms = std::vector<std::pair<int, mint>>;
	enum class Kind { inverse, logarithm, exponential, power };
	static Terms collect(const mint *f, int size, int n, int limit = 12) {
		Terms terms;
		for (int i = 1; i < std::min(size, n) && int(terms.size()) <= limit; ++i)
			if (f[i] != mint(0)) terms.emplace_back(i, f[i]);
		return terms;
	}
};
}

#line 5 "components/fps_sparse_scalar.hpp"

namespace fps_detail {
struct ScalarSparseSeries : SparseTerms {
	template <Kind kind>
	static Series get(Terms terms, int n, mint initial, mint exponent, const mint *(*inverses)(int)) {
		assert(n >= 0 && terms.size() <= 64 && (kind == Kind::inverse || unsigned(n) < mint::mod()));
		if (!n) return {};
		Series out(n);
		out[0] = initial;
		if (terms.empty()) return out;
		if constexpr (kind == Kind::power) {
			unsigned e = exponent.val();
			if (e <= 4 && terms.size() <= 12 && (n >= 4096 || terms.size() <= 3)) {
				Terms power{{0, initial}};
				for (unsigned k = 0; k < e; ++k) {
					const auto previous = power;
					for (auto [i, x] : previous) for (auto [j, y] : terms)
						if (i < n - j) power.emplace_back(i + j, x * y);
				}
				out[0] = 0;
				for (auto [i, x] : power) out[i] += x;
				return out;
			}
		}
		int stride = 0;
		for (auto [j, x] : terms) stride = std::gcd(stride, j);
		const int count = (n - 1) / stride + 1;
		for (auto &[j, x] : terms) j /= stride;
		if constexpr (kind == Kind::exponential) for (auto &[j, x] : terms) x *= mint(j);
		const mint *inv = kind == Kind::inverse ? nullptr : inverses(count);
		if (terms.size() == 1) {
			mint value = kind == Kind::logarithm ? mint(-1) : initial;
			mint ratio = kind == Kind::exponential ? terms[0].second : -terms[0].second;
			for (int i = 1; i < count; ++i) {
				value *= ratio;
				if constexpr (kind == Kind::power) value *= (mint(i - 1) - exponent) * inv[i];
				if constexpr (kind == Kind::exponential) value *= inv[i];
				out[i * stride] = value;
			}
		} else for (int i = terms[0].first; i < count; ++i) {
			mint sum = 0;
			for (auto [j, x] : terms) {
				if (j > i) break;
				if constexpr (kind == Kind::power) x *= (exponent + mint(1)) * mint(j) - mint(i);
				if constexpr (kind == Kind::inverse || kind == Kind::logarithm) x = -x;
				sum += x * out[(i - j) * stride];
				if constexpr (kind == Kind::logarithm) if (j == i) sum -= mint(j) * x;
			}
			if constexpr (kind == Kind::power || kind == Kind::exponential) sum *= inv[i];
			out[i * stride] = sum;
		}
		if constexpr (kind == Kind::logarithm) for (int i = 1; i < count; ++i) out[i * stride] *= inv[i];
		return out;
	}
};
}

#line 18 "components/fps_light_core.hpp"

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

template<class T = mint>
struct [[nodiscard, gnu::warn_unused]] LightFormalPowerSeries : std::vector<T> {
	static_assert(T::mod() == 998244353);
	using base = std::vector<T>;
	using F = LightFormalPowerSeries;
	using Sparse = fps_detail::ScalarSparseSeries;
	using Kind = Sparse::Kind;
	using Size = std::size_t;
	using base::base;
	LightFormalPowerSeries() = default;
	LightFormalPowerSeries(base a) : base(std::move(a)) {}
	F &operator=(base a) { base::operator=(std::move(a)); return *this; }
	F &operator=(std::initializer_list<T> a) { base::operator=(a); return *this; }
	static F multiply(const F &a, const F &b) {
		if constexpr (std::is_same_v<T, mint>) return fps_light_detail::multiply(a, b);
		else {
			auto x = a.native();
			return from_native(&a == &b ? Native::multiply(x, x) : Native::multiply(x, b.native()));
		}
	}
	F operator-() const { F r = *this; for (T &x : r) x = -x; return r; }
	F &operator+=(const F &g) {
		if (this->size() < g.size()) this->resize(g.size());
		for (Size i = 0; i < g.size(); ++i) (*this)[i] += g[i];
		return *this;
	}
	F &operator-=(const F &g) {
		if (this->size() < g.size()) this->resize(g.size());
		for (Size i = 0; i < g.size(); ++i) (*this)[i] -= g[i];
		return *this;
	}
	F &operator+=(T x) { if (this->empty()) this->resize(1); (*this)[0] += x; return *this; }
	F &operator-=(T x) { return *this += -x; }
	F &operator*=(const F &g) { return *this = multiply(*this, g); }
	F &operator/=(const F &g) { return *this = div_series(g); }
	F &operator%=(const F &g) { return *this = divmod(g).second; }
	F &operator*=(T x) { for (T &v : *this) v *= x; return *this; }
	F &operator/=(T x) { return *this *= x.inv(); }
	friend F operator+(F a, const F &b) { return a += b; }
	friend F operator-(F a, const F &b) { return a -= b; }
	friend F operator*(const F &a, const F &b) { return multiply(a, b); }
	friend F operator/(const F &a, const F &b) { return a.div_series(b); }
	friend F operator%(const F &a, const F &b) { return a.divmod(b).second; }
	friend F operator+(F a, T b) { return a += b; }
	friend F operator-(F a, T b) { return a -= b; }
	friend F operator*(F a, T b) { return a *= b; }
	friend F operator/(F a, T b) { return a /= b; }
	friend F operator+(T a, F b) { return b += a; }
	friend F operator-(T a, const F &b) { return -b + a; }
	friend F operator*(T a, F b) { return b *= a; }
	F &operator<<=(int d) {
		assert(d >= 0);
		auto edge = std::shift_right(this->begin(), this->end(), d);
		std::fill(this->begin(), edge, T(0));
		return *this;
	}
	F &operator>>=(int d) {
		assert(d >= 0);
		auto edge = std::shift_left(this->begin(), this->end(), d);
		std::fill(edge, this->end(), T(0));
		return *this;
	}
	friend F operator<<(F a, int d) { return a <<= d; }
	friend F operator>>(F a, int d) { return a >>= d; }

	F pre(int n) const {
		assert(n >= 0);
		return F(this->begin(), this->begin() + std::min(Size(n), this->size()));
	}
	void shrink() { while (!this->empty() && this->back() == T(0)) this->pop_back(); }
	void trim() { shrink(); }
	[[nodiscard]] T eval(T x) const {
		T res = 0;
		for (auto it = this->rbegin(); it != this->rend(); ++it) res = res * x + *it;
		return res;
	}
	F diff() const {
		F res(this->size());
		for (Size i = 1; i < this->size(); ++i) res[i - 1] = (*this)[i] * T(i);
		return res;
	}
	F integral() const {
		assert(this->size() < T::mod());
		F res(this->size());
		const auto &inv = inverses(static_cast<int>(this->size()));
		for (Size i = 1; i < this->size(); ++i) res[i] = (*this)[i - 1] * inv[i];
		return res;
	}
	void onemul(int d, T c) {
		assert(d >= 0);
		if (!d) { *this *= T(1) + c; return; }
		for (Size i = this->size(); i > Size(d); --i) (*this)[i - 1] += (*this)[i - 1 - d] * c;
	}
	void onediv(int d, T c) {
		assert(d >= 0);
		if (!d) { *this /= T(1) + c; return; }
		for (Size i = d; i < this->size(); ++i) (*this)[i] -= (*this)[i - d] * c;
	}
	F inv(int deg = -1) const {
		const int n = precision(deg);
		if (!n) return {};
		assert(!this->empty() && (*this)[0] != T(0));
		if constexpr (!std::is_same_v<T, mint>) return from_native(native().inv(n));
		else {
			if (auto r = sparse<Kind::inverse>(0, n, 32, 12, T(0)); !r.empty()) return r;
			const int work = newton_precision(n), initial = 32;
			F res = pre(initial).inv(initial);
			const T recip = res[0];
			for (int k = initial; k < work; k *= 2) inverse_step(*this, res, std::min(k * 2, work));
			res.resize(n);
			for (int i = work; i < n; ++i) res[i] = -dot(*this, res, i) * recip;
			return res;
		}
	}
	F div_series(const F &g, int deg = -1) const {
		const int n = precision(deg);
		if (!n) return {};
		assert(!g.empty() && g[0] != T(0));
		if constexpr (!std::is_same_v<T, mint>) return from_native(native().div_series(g.native(), n));
		else {
			if (n <= 32) {
				F res(n);
				const T recip = g[0].inv();
				for (int i = 0; i < n; ++i) {
					res[i] = (coefficient(i) - dot(g, res, i)) * recip;
				}
				return res;
			}
			const int work = newton_precision(n), k = std::bit_floor(unsigned(work - 1)), length = k * 2;
			F recip = g.inv(k), res = pre(k);
			const T constant = recip[0];
			recip.resize(length), res.resize(length);
			transform(recip);
			inverse_product(res, recip);
			zero(res, k);
			F quotient = res, error = g.pre(work);
			error.resize(length);
			transform(quotient);
			inverse_product(error, quotient);
			zero(error, 0, k);
			for (int i = k; i < work; ++i) error[i] = coefficient(i) - error[i];
			zero(error, work);
			inverse_product(error, recip);
			res.resize(n);
			std::copy(error.begin() + k, error.begin() + work, res.begin() + k);
			for (int i = work; i < n; ++i) {
				res[i] = (coefficient(i) - dot(g, res, i)) * constant;
			}
			return res;
		}
	}
	F log(int deg = -1) const {
		const int n = precision(deg);
		if (!n) return {};
		assert(!this->empty() && (*this)[0] == T(1));
		if constexpr (!std::is_same_v<T, mint>) return from_native(native().log(n));
		else {
			if (n >= 256) if (auto r = sparse<Kind::logarithm>(0, n, 12, 0, T(0)); !r.empty()) return r;
			F res = pre(n).diff().div_series(*this, n - 1);
			res.resize(n);
			return res.integral();
		}
	}
	F exp(int deg = -1) const {
		const int n = precision(deg);
		if (!n) return {};
		assert(this->empty() || (*this)[0] == T(0));
		if constexpr (!std::is_same_v<T, mint>) return from_native(native().exp(n));
		else {
			if (auto r = sparse<Kind::exponential>(0, n, 64, 24, T(1)); !r.empty()) return r;
			const int work = newton_precision(n), initial = 32;
			const auto &inv = inverses(n);
			F res = pre(initial).exp(initial);
			F reciprocal = res.inv(), recip;
			for (int k = initial; k < work; k *= 2) {
				const int length = k * 2, m = std::min(length, work);
				F freq = res;
				freq.resize(length);
				transform(freq);
				if (k > initial) inverse_update(freq.pre(k), recip, reciprocal, k);
				F error(k);
				for (int i = 1; i < k && Size(i) < this->size(); ++i) error[i - 1] = (*this)[i] * T(i);
				inverse_product(error, freq);
				const T last = -error.back();
				for (int i = k - 1; i; --i) error[i] = res[i] * T(i) - error[i - 1];
				error[0] = last;
				error.resize(length);
				recip = reciprocal;
				recip.resize(length);
				transform(recip);
				inverse_product(error, recip);
				for (int i = 0; i < m - k; ++i) error[i] = coefficient(k + i) - error[i] * inv[k + i];
				zero(error, m - k);
				inverse_product(error, freq);
				res.resize(m);
				std::copy_n(error.begin(), m - k, res.begin() + k);
			}
			res.resize(n);
			for (int i = work; i < n; ++i) res[i] = dot<true>(*this, res, i) * inv[i];
			return res;
		}
	}
	F sqrt(int deg = -1) const {
		const int n = precision(deg);
		if (!n) return {};
		if constexpr (!std::is_same_v<T, mint>) return from_native(native().sqrt(n));
		else {
			int first = 0, last = static_cast<int>(std::min(this->size(), Size(n)));
			while (first < last && (*this)[first] == T(0)) ++first;
			if (first == last) return F(n);
			if (first & 1) return {};
			auto constant = scalar_sqrt((*this)[first]);
			if (!constant) return {};
			const int shift = first / 2, target = n - shift, work = newton_precision(target), initial = std::min(work, 32);
			if (target >= 256) {
				F r = sparse<Kind::power>(first, target, 4, 0, *constant, T(2).inv());
				if (!r.empty()) { r.insert(r.begin(), shift, T(0)); return r; }
			}
			F f(this->begin() + first, this->begin() + last);
			const T half = T(2).inv();
			F res = f.template sparse<Kind::power>(0, initial, 32, 0, *constant, half);
			if (initial == target) {
				res.insert(res.begin(), shift, T(0));
				return res;
			}
			F reciprocal = res.inv();
			for (int k = initial; k < work; k *= 2) {
				const int length = k * 2, m = std::min(length, work);
				F error;
				if (k > initial) inverse_step(res, reciprocal, k, &error);
				if (error.empty()) { error = res; transform(error); }
				F recip = reciprocal;
				pointwise(error, error);
				transform(error, true);
				error.resize(length), recip.resize(length);
				for (int i = 0; i < m - k; ++i) {
					T low = f.coefficient(i);
					T high = f.coefficient(k + i);
					error[i] = (low + high - error[i]) * half;
				}
				zero(error, m - k);
				transform(error);
				transform(recip);
				pointwise(error, recip);
				transform(error, true);
				res.resize(m);
				std::copy_n(error.begin(), m - k, res.begin() + k);
			}
			res.resize(target);
			for (int i = work; i < target; ++i)
				res[i] = (f.coefficient(i) - dot(res, res, i, i - 1)) * reciprocal[0] * half;
			res.insert(res.begin(), shift, T(0));
			return res;
		}
	}
	F pow(long long k) const {
		assert(k >= 0);
		const int n = precision(-1);
		if (!n) return {};
		if (!k) { F res(n); res[0] = 1; return res; }
		if (k == 1) return *this;
		int first = 0;
		while (first < n && (*this)[first] == T(0)) ++first;
		if (first == n || first > (n - 1) / k) return F(n);
		const int shift = static_cast<int>(first * k), len = n - shift;
		const T scale = (*this)[first].pow(k);
		if constexpr (std::is_same_v<T, mint>) {
			F r = sparse<Kind::power>(first, len, 12, 0, scale, T(k));
			if (!r.empty()) { r.insert(r.begin(), shift, T(0)); return r; }
		}
		if (k <= 4 && first == 0) {
			F square = multiply(*this, *this).pre(n);
			return k == 2 ? square : multiply(square, k == 3 ? *this : square).pre(n);
		}
		F f(this->begin() + first, this->end());
		f.resize(len);
		f /= f[0];
		F res = (f.log(len) * T(k)).exp(len) * scale;
		res.insert(res.begin(), shift, T(0));
		return res;
	}
	[[nodiscard]] std::pair<F, F> divmod(const F &divisor) const {
		F a = *this, b = divisor;
		a.shrink(), b.shrink();
		assert(!b.empty());
		if (a.size() < b.size()) return {{}, std::move(a)};
		assert(a.size() <= (Size(1) << 22));
		const int qn = static_cast<int>(a.size() - b.size() + 1), m = static_cast<int>(b.size());
		if (m <= 32 || Size(m) * qn <= 2048) {
			F q(qn);
			const T recip = b.back().inv();
			for (int i = qn - 1; i >= 0; --i) {
				T value = q[i] = a[i + m - 1] * recip;
				for (int j = 0; j < m; ++j) a[i + j] -= value * b[j];
			}
			a.resize(m - 1);
			a.shrink(), q.shrink();
			return {std::move(q), std::move(a)};
		}
		F ra(a.rbegin(), a.rend()), rb(b.rbegin(), b.rend());
		F q = ra.div_series(rb, qn);
		std::reverse(q.begin(), q.end());
		F r = a - multiply(q, b);
		r.resize(m - 1);
		r.shrink(), q.shrink();
		return {std::move(q), std::move(r)};
	}

  private:
	template <bool Weighted = false>
	static T dot(const F &a, const F &b, int i, int last = -1) {
		T sum = 0;
		if (last < 0) last = i;
		for (int j = 1; j <= last && Size(j) < a.size(); ++j) {
			T value = a[j] * b[i - j];
			if constexpr (Weighted) value *= T(j);
			sum += value;
		}
		return sum;
	}
	T coefficient(int i) const { return Size(i) < this->size() ? (*this)[i] : T(0); }
	static void inverse_product(F &a, const F &freq) {
		transform(a); pointwise(a, freq); transform(a, true);
	}
	static void zero(F &a, int first, int last = -1) {
		std::fill(a.begin() + first, last < 0 ? a.end() : a.begin() + last, T(0));
	}
	static void transform(F &a, bool recip = false) { fps_light_detail::transform(a, recip); }
	static void pointwise(F &a, const F &b) { fps_light_detail::pointwise(a, b); }

	template <Kind kind>
	F sparse(int first, int n, int limit, int budget, T initial, T exponent = T(0)) const {
		auto terms = Sparse::collect(this->data() + first, int(this->size()) - first, n, limit);
		int stride = 0;
		for (auto [j, x] : terms) stride = std::gcd(stride, j);
		if (int(terms.size()) > limit || (budget && n > 256 && terms.size() > Size(budget * stride))) return {};
		if constexpr (kind == Kind::inverse || kind == Kind::power) {
			T recip = (*this)[first].inv();
			for (auto &[j, x] : terms) x *= recip;
			if constexpr (kind == Kind::inverse) initial = recip;
		}
		return F(Sparse::get<kind>(std::move(terms), n, initial, exponent, inverse_table));
	}
	static int newton_precision(int n) {
		const int prefix = std::bit_floor(unsigned(n));
		return prefix >= 4096 && n - prefix <= 4 ? prefix : n;
	}
	using Native = LightFormalPowerSeries<mint>;
	Native native() const {
		Native res(this->size());
		for (Size i = 0; i < this->size(); ++i) res[i] = mint::raw((*this)[i].val());
		return res;
	}
	static F from_native(const Native &a) {
		F res(a.size());
		for (Size i = 0; i < a.size(); ++i) res[i] = T::raw(a[i].val());
		return res;
	}
	static void inverse_step(const F &f, F &res, int n, F *freq = nullptr) {
		const int k = static_cast<int>(res.size()), length = k * 2;
		assert(k > 0 && std::has_single_bit(unsigned(k)) && k < n && n <= length);
		F error = f.pre(n), recip = res;
		error.resize(length), recip.resize(length);
		transform(error);
		if (freq) *freq = error;
		transform(recip);
		inverse_update(std::move(error), recip, res, n);
	}
	static void inverse_update(F error, const F &recip, F &res, int n) {
		const int k = static_cast<int>(res.size());
		pointwise(error, recip);
		transform(error, true);
		zero(error, 0, k);
		inverse_product(error, recip);
		res.resize(n);
		for (int i = k; i < n; ++i) res[i] = -error[i];
	}
	int precision(int deg) const {
		assert(this->size() <= Size(std::numeric_limits<int>::max()));
		int n = deg < 0 ? static_cast<int>(this->size()) : deg;
		assert(0 <= n && n <= (1 << 22));
		return n;
	}
	static const mint *inverse_table(int n) {
		if constexpr (std::is_same_v<T, mint>) return inverses(n).data();
		else return nullptr;
	}
	static const std::vector<T> &inverses(int n) {
		static std::vector<T> inv{T(0), T(1)};
		for (int i = static_cast<int>(inv.size()); i < n; ++i) inv.push_back(T::raw(T::mod() - T::mod() / i) * inv[T::mod() % i]);
		return inv;
	}
	static std::optional<T> scalar_sqrt(T a) {
		if (a == T(1)) return a;
		constexpr int q = 119;
		T t = a.pow((q - 1) / 2), x = t * a, c = T(3).pow(q);
		t *= x;
		int m = 23;
		while (t != T(1)) {
			int i = 0;
			for (T next = t; next != T(1) && i < m; next *= next) ++i;
			if (i == m) return std::nullopt;
			T b = c.pow(1LL << (m - i - 1));
			x *= b;
			c = b * b;
			t *= c;
			m = i;
		}
		return T::raw(std::min(x.val(), T::mod() - x.val()));
	}
};

using light_fps = LightFormalPowerSeries<>;
using LightFPS = light_fps;

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


#line 1 "components/polynomial_product_tree.hpp"



#line 1 "components/polynomial_multiply.hpp"



#line 6 "components/polynomial_multiply.hpp"

namespace polynomial_detail {
template<class F, std::size_t Limit = 1U << 23>
struct Products {
	static_assert(Limit >= 2);
	static F multiply(const F& a, const F& b) {
		if (a.empty() || b.empty()) return {};
		const auto n = a.size() + b.size() - 1;
		return n <= Limit ? F::multiply(a, b) : multiply_low(a, b, n);
	}
	static F multiply_low(const F& a, const F& b, std::size_t n) {
		F result(n);
		constexpr std::size_t block = Limit / 2;
		for (std::size_t i = 0; i < std::min(a.size(), n); i += block)
			for (std::size_t j = 0; j < std::min(b.size(), n - i); j += block) {
				F x(a.begin() + i, a.begin() + std::min(i + block, a.size()));
				F y(b.begin() + j, b.begin() + std::min(j + block, b.size()));
				F p = F::multiply(x, y);
				for (std::size_t k = 0; k < std::min(p.size(), n - i - j); ++k)
					result[i + j + k] += p[k];
			}
		return result;
	}
};
}


#line 10 "components/polynomial_product_tree.hpp"

namespace polynomial_detail {
template<class F, std::size_t Limit = 1U << 23>
struct ProductTree : Products<F, Limit> {
	using Products<F, Limit>::multiply;
	using Products<F, Limit>::multiply_low;
	static_assert(Limit >= 2);
	using T = typename F::value_type;
	std::size_t size = 1, count;
	std::vector<F> product;
	static void trim(F& f) { while (!f.empty() && f.back() == T(0)) f.pop_back(); }


	explicit ProductTree(const std::vector<T>& xs) : count(xs.size()) {
		assert(count <= Limit);
		while (size < count) size <<= 1;
		product.resize(2 * size);
		for (std::size_t i = 0; i < size; ++i)
			product[size + i] = i < count ? F{-xs[i], T(1)} : F{T(1)};
		for (std::size_t i = size; --i;) {
			const auto& a = product[2 * i];
			const auto& b = product[2 * i + 1];
			product[i] = b.size() == 1 ? a : multiply(a, b);
		}
	}

	std::vector<T> evaluate(const F& f) const {
		std::vector<T> result(count);
		if (!count) return result;
		auto remainder = [](const F& a, const F& b) {
			return a.size() < b.size() ? a : a.divmod(b).second;
		};
		auto visit = [&](auto&& self, std::size_t node, const F& value) -> void {
			if (value.empty()) return;
			if (node >= size) {
				result[node - size] = value[0];
				return;
			}
			self(self, node * 2, remainder(value, product[node * 2]));
			if (product[node * 2 + 1].size() > 1)
				self(self, node * 2 + 1, remainder(value, product[node * 2 + 1]));
		};
		F value = f;
		trim(value);
		visit(visit, 1, remainder(value, product[1]));
		return result;
	}

	F interpolate(const std::vector<T>& ys) const {
		assert(ys.size() == count);
		if (!count) return {};
		F derivative = product[1].diff();
		trim(derivative);
		std::vector<T> denominator = evaluate(derivative), prefix(count + 1, T(1));
		for (std::size_t i = 0; i < count; ++i) {
			assert(denominator[i] != T(0));
			prefix[i + 1] = prefix[i] * denominator[i];
		}
		T inverse = prefix[count].inv();
		std::vector<F> value(2 * size);
		for (std::size_t i = count; i--;) {
			value[size + i] = F{ys[i] * inverse * prefix[i]};
			inverse *= denominator[i];
		}
		for (std::size_t i = size; --i;) {
			value[i] = F::multiply(value[2 * i], product[2 * i + 1])
				+ F::multiply(value[2 * i + 1], product[2 * i]);
			value[2 * i] = F();
			value[2 * i + 1] = F();
		}
		value[1].resize(count);
		return std::move(value[1]);
	}
};
}


#line 6 "components/fps_light_backend.hpp"

namespace fps_light_detail {
template<class T>
using ProductTree = polynomial_detail::ProductTree<LightFormalPowerSeries<T>>;
}

namespace light_fps_algorithms {
template<class T>
LightFormalPowerSeries<T> taylor_shift(LightFormalPowerSeries<T> f, std::type_identity_t<T> c) {
	using F = LightFormalPowerSeries<T>;
	const std::size_t n = f.size();
	if (n <= 1 || c == T(0)) return f;
	assert(n < T::mod());
	F a(n), b(n);
	T factorial = 1;
	a[n - 1] = f[0];
	for (std::size_t i = 1; i < n; ++i) {
		factorial *= T(i);
		a[n - 1 - i] = f[i] * factorial;
	}
	f[n - 1] = factorial.inv();
	for (std::size_t i = n - 1; i; --i) f[i - 1] = f[i] * T(i);
	T power = 1;
	for (std::size_t i = 0; i < n; ++i) {
		b[i] = power * f[i];
		power *= c;
	}
	a = F::multiply(a, b);
	for (std::size_t i = 0; i < n; ++i) f[i] *= a[n - 1 - i];
	return f;
}

template<class T>
T bostan_mori(LightFormalPowerSeries<T> p, LightFormalPowerSeries<T> q, long long k) {
	using F = LightFormalPowerSeries<T>;
	assert(k >= 0 && !q.empty() && q[0] != T(0));
	p.shrink();
	q.shrink();
	while (k && !p.empty() && q.size() > 1) {
		F even((q.size() + 1) / 2), odd(q.size() / 2);
		for (std::size_t i = 0; i < q.size(); ++i) (i & 1 ? odd : even)[i / 2] = q[i];
		for (std::size_t i = 1; i < q.size(); i += 2) q[i] = -q[i];
		F numerator = F::multiply(p, q);
		q = F::multiply(even, even), odd = F::multiply(odd, odd);
		p.resize((numerator.size() + 1 - std::size_t(k & 1)) / 2);
		for (std::size_t i = 0; i < p.size(); ++i) p[i] = numerator[2 * i + std::size_t(k & 1)];
		if (q.size() == odd.size()) q.push_back(T(0));
		for (std::size_t i = 0; i < odd.size(); ++i) q[i + 1] -= odd[i];
		p.shrink();
		k >>= 1;
	}
	return static_cast<unsigned long long>(k) < p.size() ? p[std::size_t(k)] / q[0] : T(0);
}

template<class T>
T linear_recurrence(const std::vector<T> &initial, const std::vector<T> &coefficients, long long k) {
	using F = LightFormalPowerSeries<T>;
	assert(k >= 0 && initial.size() == coefficients.size());
	const std::size_t n = initial.size();
	if (static_cast<unsigned long long>(k) < n) return initial[std::size_t(k)];
	if (!n) return T(0);
	F q(n + 1);
	q[0] = T(1);
	for (std::size_t i = 0; i < n; ++i) q[i + 1] = -coefficients[i];
	F p = F::multiply(F(initial.begin(), initial.end()), q).pre(static_cast<int>(n));
	return bostan_mori(std::move(p), std::move(q), k);
}

template<class T>
std::vector<T> multipoint_eval(const LightFormalPowerSeries<T> &f, const std::vector<T> &xs) {
	if (f.size() <= 32 || xs.size() <= 32) {
		std::vector<T> result;
		result.reserve(xs.size());
		for (T x : xs) result.push_back(f.eval(x));
		return result;
	}
	return fps_light_detail::ProductTree<T>(xs).evaluate(f);
}

template<class T>
LightFormalPowerSeries<T> interpolate(const std::vector<T> &xs, const std::vector<T> &ys) {
	assert(xs.size() == ys.size());
	return fps_light_detail::ProductTree<T>(xs).interpolate(ys);
}
}


#line 4 "fps/fps_light.hpp"

#line 1 "components/fast_io.hpp"



#line 5 "components/fast_io.hpp"
#include <array>
#line 8 "components/fast_io.hpp"
#include <charconv>
#line 10 "components/fast_io.hpp"
#include <cstdint>
#include <cstdio>
#include <cstring>
#include <memory>
#include <stdexcept>
#include <string>
#include <string_view>
#line 18 "components/fast_io.hpp"
#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 6 "test/fixtures/fps_light_io.hpp"

inline light_fps read_light_fps(FastInput &input, int n) {
	light_fps result(n);
	for (auto &x : result) { unsigned value = 0; input.read(value); x = mint::raw(value); }
	return result;
}

inline void write_light_fps(FastOutput &output, const light_fps &f) {
	for (auto x : f) { output.write(x.val()); output.write(' '); }
	output.write('\n');
}


#line 3 "test/fps_light_sqrt.test.cpp"

int main() {
	FastInput input;
	FastOutput output;
	int n = 0; input.read(n);
	auto root = read_light_fps(input, n).sqrt();
	if (root.empty()) output.println(-1);
	else write_light_fps(output, root);
}