結果
| 問題 |
No.8024 等式
|
| コンテスト | |
| ユーザー |
 NyaanNyaan
|
| 提出日時 | 2023-09-12 15:45:55 |
| 言語 | C++17 (gcc 13.3.0 + boost 1.87.0) |
| 結果 |
WA
(最新)
AC
(最初)
|
| 実行時間 | - |
| コード長 | 53,015 bytes |
| コンパイル時間 | 5,266 ms |
| コンパイル使用メモリ | 335,388 KB |
| 最終ジャッジ日時 | 2025-02-16 22:02:31 |
|
ジャッジサーバーID （参考情報） |
judge5 / judge2 |
（要ログイン）
| ファイルパターン | 結果 |
|---|---|
| other | AC * 22 WA * 1 |
ソースコード
/**
* date : 2023-09-12 15:45:41
* author : Nyaan
*/
#define NDEBUG
using namespace std;
// intrinstic
#include <immintrin.h>
#include <algorithm>
#include <array>
#include <bitset>
#include <cassert>
#include <cctype>
#include <cfenv>
#include <cfloat>
#include <chrono>
#include <cinttypes>
#include <climits>
#include <cmath>
#include <complex>
#include <cstdarg>
#include <cstddef>
#include <cstdint>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <deque>
#include <fstream>
#include <functional>
#include <initializer_list>
#include <iomanip>
#include <ios>
#include <iostream>
#include <istream>
#include <iterator>
#include <limits>
#include <list>
#include <map>
#include <memory>
#include <new>
#include <numeric>
#include <ostream>
#include <queue>
#include <random>
#include <set>
#include <sstream>
#include <stack>
#include <streambuf>
#include <string>
#include <tuple>
#include <type_traits>
#include <typeinfo>
#include <unordered_map>
#include <unordered_set>
#include <utility>
#include <vector>
// utility
namespace Nyaan {
using ll = long long;
using i64 = long long;
using u64 = unsigned long long;
using i128 = __int128_t;
using u128 = __uint128_t;
template <typename T>
using V = vector<T>;
template <typename T>
using VV = vector<vector<T>>;
using vi = vector<int>;
using vl = vector<long long>;
using vd = V<double>;
using vs = V<string>;
using vvi = vector<vector<int>>;
using vvl = vector<vector<long long>>;
template <typename T>
using minpq = priority_queue<T, vector<T>, greater<T>>;
template <typename T, typename U>
struct P : pair<T, U> {
template <typename... Args>
P(Args... args) : pair<T, U>(args...) {}
using pair<T, U>::first;
using pair<T, U>::second;
P &operator+=(const P &r) {
first += r.first;
second += r.second;
return *this;
}
P &operator-=(const P &r) {
first -= r.first;
second -= r.second;
return *this;
}
P &operator*=(const P &r) {
first *= r.first;
second *= r.second;
return *this;
}
template <typename S>
P &operator*=(const S &r) {
first *= r, second *= r;
return *this;
}
P operator+(const P &r) const { return P(*this) += r; }
P operator-(const P &r) const { return P(*this) -= r; }
P operator*(const P &r) const { return P(*this) *= r; }
template <typename S>
P operator*(const S &r) const {
return P(*this) *= r;
}
P operator-() const { return P{-first, -second}; }
};
using pl = P<ll, ll>;
using pi = P<int, int>;
using vp = V<pl>;
constexpr int inf = 1001001001;
constexpr long long infLL = 4004004004004004004LL;
template <typename T>
int sz(const T &t) {
return t.size();
}
template <typename T, typename U>
inline bool amin(T &x, U y) {
return (y < x) ? (x = y, true) : false;
}
template <typename T, typename U>
inline bool amax(T &x, U y) {
return (x < y) ? (x = y, true) : false;
}
template <typename T>
inline T Max(const vector<T> &v) {
return *max_element(begin(v), end(v));
}
template <typename T>
inline T Min(const vector<T> &v) {
return *min_element(begin(v), end(v));
}
template <typename T>
inline long long Sum(const vector<T> &v) {
return accumulate(begin(v), end(v), 0LL);
}
template <typename T>
int lb(const vector<T> &v, const T &a) {
return lower_bound(begin(v), end(v), a) - begin(v);
}
template <typename T>
int ub(const vector<T> &v, const T &a) {
return upper_bound(begin(v), end(v), a) - begin(v);
}
constexpr long long TEN(int n) {
long long ret = 1, x = 10;
for (; n; x *= x, n >>= 1) ret *= (n & 1 ? x : 1);
return ret;
}
template <typename T, typename U>
pair<T, U> mkp(const T &t, const U &u) {
return make_pair(t, u);
}
template <typename T>
vector<T> mkrui(const vector<T> &v, bool rev = false) {
vector<T> ret(v.size() + 1);
if (rev) {
for (int i = int(v.size()) - 1; i >= 0; i--) ret[i] = v[i] + ret[i + 1];
} else {
for (int i = 0; i < int(v.size()); i++) ret[i + 1] = ret[i] + v[i];
}
return ret;
};
template <typename T>
vector<T> mkuni(const vector<T> &v) {
vector<T> ret(v);
sort(ret.begin(), ret.end());
ret.erase(unique(ret.begin(), ret.end()), ret.end());
return ret;
}
template <typename F>
vector<int> mkord(int N, F f) {
vector<int> ord(N);
iota(begin(ord), end(ord), 0);
sort(begin(ord), end(ord), f);
return ord;
}
template <typename T>
vector<int> mkinv(vector<T> &v) {
int max_val = *max_element(begin(v), end(v));
vector<int> inv(max_val + 1, -1);
for (int i = 0; i < (int)v.size(); i++) inv[v[i]] = i;
return inv;
}
vector<int> mkiota(int n) {
vector<int> ret(n);
iota(begin(ret), end(ret), 0);
return ret;
}
template <typename T>
T mkrev(const T &v) {
T w{v};
reverse(begin(w), end(w));
return w;
}
template <typename T>
bool nxp(vector<T> &v) {
return next_permutation(begin(v), end(v));
}
// 返り値の型は入力の T に依存
// i 要素目 : [0, a[i])
template <typename T>
vector<vector<T>> product(const vector<T> &a) {
vector<vector<T>> ret;
vector<T> v;
auto dfs = [&](auto rc, int i) -> void {
if (i == (int)a.size()) {
ret.push_back(v);
return;
}
for (int j = 0; j < a[i]; j++) v.push_back(j), rc(rc, i + 1), v.pop_back();
};
dfs(dfs, 0);
return ret;
}
// F : function(void(T&)), mod を取る操作
// T : 整数型のときはオーバーフローに注意する
template <typename T>
T Power(T a, long long n, const T &I, const function<void(T &)> &f) {
T res = I;
for (; n; f(a = a * a), n >>= 1) {
if (n & 1) f(res = res * a);
}
return res;
}
// T : 整数型のときはオーバーフローに注意する
template <typename T>
T Power(T a, long long n, const T &I) {
return Power(a, n, I, function<void(T &)>{[](T &) -> void {}});
}
} // namespace Nyaan
// bit operation
namespace Nyaan {
__attribute__((target("popcnt"))) inline int popcnt(const u64 &a) {
return _mm_popcnt_u64(a);
}
inline int lsb(const u64 &a) { return a ? __builtin_ctzll(a) : 64; }
inline int ctz(const u64 &a) { return a ? __builtin_ctzll(a) : 64; }
inline int msb(const u64 &a) { return a ? 63 - __builtin_clzll(a) : -1; }
template <typename T>
inline int gbit(const T &a, int i) {
return (a >> i) & 1;
}
template <typename T>
inline void sbit(T &a, int i, bool b) {
if (gbit(a, i) != b) a ^= T(1) << i;
}
constexpr long long PW(int n) { return 1LL << n; }
constexpr long long MSK(int n) { return (1LL << n) - 1; }
} // namespace Nyaan
// inout
namespace Nyaan {
template <typename T, typename U>
ostream &operator<<(ostream &os, const pair<T, U> &p) {
os << p.first << " " << p.second;
return os;
}
template <typename T, typename U>
istream &operator>>(istream &is, pair<T, U> &p) {
is >> p.first >> p.second;
return is;
}
template <typename T>
ostream &operator<<(ostream &os, const vector<T> &v) {
int s = (int)v.size();
for (int i = 0; i < s; i++) os << (i ? " " : "") << v[i];
return os;
}
template <typename T>
istream &operator>>(istream &is, vector<T> &v) {
for (auto &x : v) is >> x;
return is;
}
istream &operator>>(istream &is, __int128_t &x) {
string S;
is >> S;
x = 0;
int flag = 0;
for (auto &c : S) {
if (c == '-') {
flag = true;
continue;
}
x *= 10;
x += c - '0';
}
if (flag) x = -x;
return is;
}
istream &operator>>(istream &is, __uint128_t &x) {
string S;
is >> S;
x = 0;
for (auto &c : S) {
x *= 10;
x += c - '0';
}
return is;
}
ostream &operator<<(ostream &os, __int128_t x) {
if (x == 0) return os << 0;
if (x < 0) os << '-', x = -x;
string S;
while (x) S.push_back('0' + x % 10), x /= 10;
reverse(begin(S), end(S));
return os << S;
}
ostream &operator<<(ostream &os, __uint128_t x) {
if (x == 0) return os << 0;
string S;
while (x) S.push_back('0' + x % 10), x /= 10;
reverse(begin(S), end(S));
return os << S;
}
void in() {}
template <typename T, class... U>
void in(T &t, U &...u) {
cin >> t;
in(u...);
}
void out() { cout << "\n"; }
template <typename T, class... U, char sep = ' '>
void out(const T &t, const U &...u) {
cout << t;
if (sizeof...(u)) cout << sep;
out(u...);
}
struct IoSetupNya {
IoSetupNya() {
cin.tie(nullptr);
ios::sync_with_stdio(false);
cout << fixed << setprecision(15);
cerr << fixed << setprecision(7);
}
} iosetupnya;
} // namespace Nyaan
// debug
#ifdef NyaanDebug
#define trc(...) (void(0))
#else
#define trc(...) (void(0))
#endif
#ifdef NyaanLocal
#define trc2(...) (void(0))
#else
#define trc2(...) (void(0))
#endif
// macro
#define each(x, v) for (auto&& x : v)
#define each2(x, y, v) for (auto&& [x, y] : v)
#define all(v) (v).begin(), (v).end()
#define rep(i, N) for (long long i = 0; i < (long long)(N); i++)
#define repr(i, N) for (long long i = (long long)(N)-1; i >= 0; i--)
#define rep1(i, N) for (long long i = 1; i <= (long long)(N); i++)
#define repr1(i, N) for (long long i = (N); (long long)(i) > 0; i--)
#define reg(i, a, b) for (long long i = (a); i < (b); i++)
#define regr(i, a, b) for (long long i = (b)-1; i >= (a); i--)
#define fi first
#define se second
#define ini(...) \
int __VA_ARGS__; \
in(__VA_ARGS__)
#define inl(...) \
long long __VA_ARGS__; \
in(__VA_ARGS__)
#define ins(...) \
string __VA_ARGS__; \
in(__VA_ARGS__)
#define in2(s, t) \
for (int i = 0; i < (int)s.size(); i++) { \
in(s[i], t[i]); \
}
#define in3(s, t, u) \
for (int i = 0; i < (int)s.size(); i++) { \
in(s[i], t[i], u[i]); \
}
#define in4(s, t, u, v) \
for (int i = 0; i < (int)s.size(); i++) { \
in(s[i], t[i], u[i], v[i]); \
}
#define die(...) \
do { \
Nyaan::out(__VA_ARGS__); \
return; \
} while (0)
namespace Nyaan {
void solve();
}
int main() { Nyaan::solve(); }
//
using namespace std;
using namespace std;
namespace internal {
template <typename T>
using is_broadly_integral =
typename conditional_t<is_integral_v<T> || is_same_v<T, __int128_t> ||
is_same_v<T, __uint128_t>,
true_type, false_type>::type;
template <typename T>
using is_broadly_signed =
typename conditional_t<is_signed_v<T> || is_same_v<T, __int128_t>,
true_type, false_type>::type;
template <typename T>
using is_broadly_unsigned =
typename conditional_t<is_unsigned_v<T> || is_same_v<T, __uint128_t>,
true_type, false_type>::type;
#define ENABLE_VALUE(x) \
template <typename T> \
constexpr bool x##_v = x<T>::value;
ENABLE_VALUE(is_broadly_integral);
ENABLE_VALUE(is_broadly_signed);
ENABLE_VALUE(is_broadly_unsigned);
#undef ENABLE_VALUE
#define ENABLE_HAS_TYPE(var) \
template <class, class = void> \
struct has_##var : false_type {}; \
template <class T> \
struct has_##var<T, void_t<typename T::var>> : true_type {}; \
template <class T> \
constexpr auto has_##var##_v = has_##var<T>::value;
#define ENABLE_HAS_VAR(var) \
template <class, class = void> \
struct has_##var : false_type {}; \
template <class T> \
struct has_##var<T, void_t<decltype(T::var)>> : true_type {}; \
template <class T> \
constexpr auto has_##var##_v = has_##var<T>::value;
} // namespace internal
using namespace std;
namespace BinaryGCDImpl {
using u64 = unsigned long long;
using i8 = char;
u64 binary_gcd(u64 a, u64 b) {
if (a == 0 || b == 0) return a + b;
i8 n = __builtin_ctzll(a);
i8 m = __builtin_ctzll(b);
a >>= n;
b >>= m;
n = min(n, m);
while (a != b) {
u64 d = a - b;
i8 s = __builtin_ctzll(d);
bool f = a > b;
b = f ? b : a;
a = (f ? d : -d) >> s;
}
return a << n;
}
using u128 = __uint128_t;
// a > 0
int ctz128(u128 a) {
u64 lo = a & u64(-1);
return lo ? __builtin_ctzll(lo) : 64 + __builtin_ctzll(a >> 64);
}
u128 binary_gcd128(u128 a, u128 b) {
if (a == 0 || b == 0) return a + b;
i8 n = ctz128(a);
i8 m = ctz128(b);
a >>= n;
b >>= m;
n = min(n, m);
while (a != b) {
u128 d = a - b;
i8 s = ctz128(d);
bool f = a > b;
b = f ? b : a;
a = (f ? d : -d) >> s;
}
return a << n;
}
} // namespace BinaryGCDImpl
long long binary_gcd(long long a, long long b) {
return BinaryGCDImpl::binary_gcd(abs(a), abs(b));
}
__int128_t binary_gcd128(__int128_t a, __int128_t b) {
if (a < 0) a = -a;
if (b < 0) b = -b;
return BinaryGCDImpl::binary_gcd128(a, b);
}
/**
* @brief binary GCD
*/
// T : 値, U : 比較用
template <typename T, typename U>
struct RationalBase {
using R = RationalBase;
using Key = T;
T x, y;
RationalBase() : x(0), y(1) {}
template <typename T1>
RationalBase(const T1& _x) : RationalBase<T, U>(_x, T1{1}) {}
template <typename T1, typename T2>
RationalBase(const T1& _x, const T2& _y) : x(_x), y(_y) {
assert(y != 0);
if (y == -1) x = -x, y = -y;
if (y != 1) {
T g;
if constexpr (internal::is_broadly_integral_v<T>) {
if constexpr (sizeof(T) == 16) {
g = binary_gcd128(x, y);
} else {
g = binary_gcd(x, y);
}
} else {
g = gcd(x, y);
}
if (g != 0) x /= g, y /= g;
if (y < 0) x = -x, y = -y;
}
}
// y = 0 の代入も認める
static R raw(T _x, T _y) {
R r;
r.x = _x, r.y = _y;
return r;
}
friend R operator+(const R& l, const R& r) {
if (l.y == r.y) return R{l.x + r.x, l.y};
return R{l.x * r.y + l.y * r.x, l.y * r.y};
}
friend R operator-(const R& l, const R& r) {
if (l.y == r.y) return R{l.x - r.x, l.y};
return R{l.x * r.y - l.y * r.x, l.y * r.y};
}
friend R operator*(const R& l, const R& r) { return R{l.x * r.x, l.y * r.y}; }
friend R operator/(const R& l, const R& r) { return R{l.x * r.y, l.y * r.x}; }
R& operator+=(const R& r) { return (*this) = (*this) + r; }
R& operator-=(const R& r) { return (*this) = (*this) - r; }
R& operator*=(const R& r) { return (*this) = (*this) * r; }
R& operator/=(const R& r) { return (*this) = (*this) / r; }
R operator-() const { return raw(-x, y); }
R inverse() const {
assert(x != 0);
R r = raw(y, x);
if (r.y < 0) r.x = -r.x, r.y = -r.y;
return r;
}
R pow(long long p) const {
R res{1}, base{*this};
while (p) {
if (p & 1) res *= base;
base *= base;
p >>= 1;
}
return res;
}
friend bool operator==(const R& l, const R& r) {
return l.x == r.x && l.y == r.y;
};
friend bool operator!=(const R& l, const R& r) {
return l.x != r.x || l.y != r.y;
};
friend bool operator<(const R& l, const R& r) {
return U{l.x} * r.y < U{l.y} * r.x;
};
friend bool operator<=(const R& l, const R& r) { return l < r || l == r; }
friend bool operator>(const R& l, const R& r) {
return U{l.x} * r.y > U{l.y} * r.x;
};
friend bool operator>=(const R& l, const R& r) { return l > r || l == r; }
friend ostream& operator<<(ostream& os, const R& r) {
os << r.x;
if (r.x != 0 && r.y != 1) os << "/" << r.y;
return os;
}
// T にキャストされるので T が bigint の場合は to_ll も要る
T to_mint(T mod) const {
assert(mod != 0);
T a = y, b = mod, u = 1, v = 0, t;
while (b > 0) {
t = a / b;
swap(a -= t * b, b);
swap(u -= t * v, v);
}
return U((u % mod + mod) % mod) * x % mod;
}
};
using Rational = RationalBase<long long, __int128_t>;
template <typename R = Rational>
struct Binomial_rational {
vector<R> fc;
Binomial_rational(int = 0) { fc.emplace_back(1); }
void extend() {
int n = fc.size();
R nxt = fc.back() * n;
fc.push_back(nxt);
}
R fac(int n) {
if (n < 0) return 0;
while ((int)fc.size() <= n) extend();
return fc[n];
}
R finv(int n) {
if (n < 0) return 0;
return fac(n).inverse();
}
R inv(int n) {
if (n < 0) return -inv(-n);
return R{1, max(n, 1)};
}
R C(int n, int r) {
if (n < 0 or r < 0 or n < r) return R{0};
return fac(n) * finv(n - r) * finv(r);
}
R operator()(int n, int r) { return C(n, r); }
template <typename I>
R multinomial(const vector<I>& r) {
static_assert(is_integral<I>::value == true);
int n = 0;
for (auto& x : r) {
if (x < 0) return R{0};
n += x;
}
R res = fac(n);
for (auto& x : r) res *= finv(x);
return res;
}
template <typename I>
R operator()(const vector<I>& r) {
return multinomial(r);
}
};
template <typename R = Rational>
struct FormalPowerSeries_rational : vector<R> {
using vector<R>::vector;
using fps = FormalPowerSeries_rational;
fps &operator+=(const fps &r) {
if (r.size() > this->size()) this->resize(r.size());
for (int i = 0; i < (int)r.size(); i++) (*this)[i] += r[i];
return *this;
}
fps &operator+=(const R &r) {
if (this->empty()) this->resize(1);
(*this)[0] += r;
return *this;
}
fps &operator-=(const fps &r) {
if (r.size() > this->size()) this->resize(r.size());
for (int i = 0; i < (int)r.size(); i++) (*this)[i] -= r[i];
return *this;
}
fps &operator-=(const R &r) {
if (this->empty()) this->resize(1);
(*this)[0] -= r;
return *this;
}
fps &operator*=(const fps &r) {
int n = this->size() + r.size() - 1;
fps f(n);
for (int i = 0; i < (int)this->size(); i++) {
for (int j = 0; j < (int)r.size(); j++) {
f[i + j] += (*this)[i] * r[j];
}
}
return *this = f;
}
fps &operator*=(const R &v) {
for (int k = 0; k < (int)this->size(); k++) (*this)[k] *= v;
return *this;
}
fps &operator/=(const fps &r) {
if (this->size() < r.size()) {
this->clear();
return *this;
}
int n = this->size() - r.size() + 1;
fps f(*this), g(r);
g.shrink();
R coeff = g.back().inverse();
for (auto &x : g) x *= coeff;
int deg = (int)f.size() - (int)g.size() + 1;
int gs = g.size();
fps quo(deg);
for (int i = deg - 1; i >= 0; i--) {
quo[i] = f[i + gs - 1];
for (int j = 0; j < gs; j++) f[i + j] -= quo[i] * g[j];
}
*this = quo * coeff;
this->resize(n, R(0));
return *this;
}
fps &operator%=(const fps &r) {
*this -= *this / r * r;
shrink();
return *this;
}
fps operator+(const fps &r) const { return fps(*this) += r; }
fps operator+(const R &v) const { return fps(*this) += v; }
fps operator-(const fps &r) const { return fps(*this) -= r; }
fps operator-(const R &v) const { return fps(*this) -= v; }
fps operator*(const fps &r) const { return fps(*this) *= r; }
fps operator*(const R &v) const { return fps(*this) *= v; }
fps operator/(const fps &r) const { return fps(*this) /= r; }
fps operator%(const fps &r) const { return fps(*this) %= r; }
fps operator-() const {
fps ret(this->size());
for (int i = 0; i < (int)this->size(); i++) ret[i] = -(*this)[i];
return ret;
}
void shrink() {
while (this->size() && this->back() == R(0)) this->pop_back();
}
fps rev() const {
fps ret(*this);
reverse(begin(ret), end(ret));
return ret;
}
fps dot(fps r) const {
fps ret(min(this->size(), r.size()));
for (int i = 0; i < (int)ret.size(); i++) ret[i] = (*this)[i] * r[i];
return ret;
}
// 前 sz 項を取ってくる。sz に足りない項は 0 埋めする
fps pre(int sz) const {
fps ret(begin(*this), begin(*this) + min((int)this->size(), sz));
if ((int)ret.size() < sz) ret.resize(sz);
return ret;
}
fps operator>>(int sz) const {
if ((int)this->size() <= sz) return {};
fps ret(*this);
ret.erase(ret.begin(), ret.begin() + sz);
return ret;
}
fps operator<<(int sz) const {
fps ret(*this);
ret.insert(ret.begin(), sz, R(0));
return ret;
}
fps diff() const {
const int n = (int)this->size();
fps ret(max(0, n - 1));
R one(1), coeff(1);
for (int i = 1; i < n; i++) {
ret[i - 1] = (*this)[i] * coeff;
coeff += one;
}
return ret;
}
fps integral() const {
const int n = (int)this->size();
fps ret(n + 1);
for (int i = 0; i < n; i++) ret[i + 1] = (*this)[i] / (i + 1);
return ret;
}
R eval(R x) const {
R r = 0, w = 1;
for (auto &v : *this) r += w * v, w *= x;
return r;
}
fps inv(int deg = -1) const {
assert((*this)[0] != R(0));
if (deg == -1) deg = (*this).size();
fps ret{R(1) / (*this)[0]};
for (int i = 1; i < deg; i <<= 1) {
ret = (ret + ret - ret * ret * (*this).pre(i << 1)).pre(i << 1);
}
return ret.pre(deg);
}
fps log(int deg = -1) const {
assert(!(*this).empty() && (*this)[0] == R(1));
if (deg == -1) deg = (int)this->size();
return (this->diff() * this->inv(deg)).pre(deg - 1).integral();
}
fps exp(int deg = -1) const {
assert((*this).size() == 0 || (*this)[0] == R(0));
if (deg == -1) deg = (int)this->size();
fps ret{R(1)};
for (int i = 1; i < deg; i <<= 1) {
ret = (ret * (pre(i << 1) + R(1) - ret.log(i << 1))).pre(i << 1);
}
return ret.pre(deg);
}
fps pow(int64_t k, int deg = -1) const {
const int n = (int)this->size();
if (deg == -1) deg = n;
if (k == 0) {
fps ret(deg);
if (deg) ret[0] = 1;
return ret;
}
for (int i = 0; i < n; i++) {
if ((*this)[i] != R(0)) {
R rev = R(1) / (*this)[i];
fps ret = (((*this * rev) >> i).log(deg) * k).exp(deg);
ret *= (*this)[i].pow(k);
ret = (ret << (i * k)).pre(deg);
if ((int)ret.size() < deg) ret.resize(deg, R(0));
return ret;
}
if (__int128_t(i + 1) * k >= deg) return fps(deg, R(0));
}
return fps(deg, R(0));
}
};
//
using namespace std;
using namespace std;
template <uint32_t mod>
struct LazyMontgomeryModInt {
using mint = LazyMontgomeryModInt;
using i32 = int32_t;
using u32 = uint32_t;
using u64 = uint64_t;
static constexpr u32 get_r() {
u32 ret = mod;
for (i32 i = 0; i < 4; ++i) ret *= 2 - mod * ret;
return ret;
}
static constexpr u32 r = get_r();
static constexpr u32 n2 = -u64(mod) % mod;
static_assert(mod < (1 << 30), "invalid, mod >= 2 ^ 30");
static_assert((mod & 1) == 1, "invalid, mod % 2 == 0");
static_assert(r * mod == 1, "this code has bugs.");
u32 a;
constexpr LazyMontgomeryModInt() : a(0) {}
constexpr LazyMontgomeryModInt(const int64_t &b)
: a(reduce(u64(b % mod + mod) * n2)){};
static constexpr u32 reduce(const u64 &b) {
return (b + u64(u32(b) * u32(-r)) * mod) >> 32;
}
constexpr mint &operator+=(const mint &b) {
if (i32(a += b.a - 2 * mod) < 0) a += 2 * mod;
return *this;
}
constexpr mint &operator-=(const mint &b) {
if (i32(a -= b.a) < 0) a += 2 * mod;
return *this;
}
constexpr mint &operator*=(const mint &b) {
a = reduce(u64(a) * b.a);
return *this;
}
constexpr mint &operator/=(const mint &b) {
*this *= b.inverse();
return *this;
}
constexpr mint operator+(const mint &b) const { return mint(*this) += b; }
constexpr mint operator-(const mint &b) const { return mint(*this) -= b; }
constexpr mint operator*(const mint &b) const { return mint(*this) *= b; }
constexpr mint operator/(const mint &b) const { return mint(*this) /= b; }
constexpr bool operator==(const mint &b) const {
return (a >= mod ? a - mod : a) == (b.a >= mod ? b.a - mod : b.a);
}
constexpr bool operator!=(const mint &b) const {
return (a >= mod ? a - mod : a) != (b.a >= mod ? b.a - mod : b.a);
}
constexpr mint operator-() const { return mint() - mint(*this); }
constexpr mint operator+() const { return mint(*this); }
constexpr mint pow(u64 n) const {
mint ret(1), mul(*this);
while (n > 0) {
if (n & 1) ret *= mul;
mul *= mul;
n >>= 1;
}
return ret;
}
constexpr mint inverse() const {
int x = get(), y = mod, u = 1, v = 0, t = 0, tmp = 0;
while (y > 0) {
t = x / y;
x -= t * y, u -= t * v;
tmp = x, x = y, y = tmp;
tmp = u, u = v, v = tmp;
}
return mint{u};
}
friend ostream &operator<<(ostream &os, const mint &b) {
return os << b.get();
}
friend istream &operator>>(istream &is, mint &b) {
int64_t t;
is >> t;
b = LazyMontgomeryModInt<mod>(t);
return (is);
}
constexpr u32 get() const {
u32 ret = reduce(a);
return ret >= mod ? ret - mod : ret;
}
static constexpr u32 get_mod() { return mod; }
};
template <typename mint>
struct NTT {
static constexpr uint32_t get_pr() {
uint32_t _mod = mint::get_mod();
using u64 = uint64_t;
u64 ds[32] = {};
int idx = 0;
u64 m = _mod - 1;
for (u64 i = 2; i * i <= m; ++i) {
if (m % i == 0) {
ds[idx++] = i;
while (m % i == 0) m /= i;
}
}
if (m != 1) ds[idx++] = m;
uint32_t _pr = 2;
while (1) {
int flg = 1;
for (int i = 0; i < idx; ++i) {
u64 a = _pr, b = (_mod - 1) / ds[i], r = 1;
while (b) {
if (b & 1) r = r * a % _mod;
a = a * a % _mod;
b >>= 1;
}
if (r == 1) {
flg = 0;
break;
}
}
if (flg == 1) break;
++_pr;
}
return _pr;
};
static constexpr uint32_t mod = mint::get_mod();
static constexpr uint32_t pr = get_pr();
static constexpr int level = __builtin_ctzll(mod - 1);
mint dw[level], dy[level];
void setwy(int k) {
mint w[level], y[level];
w[k - 1] = mint(pr).pow((mod - 1) / (1 << k));
y[k - 1] = w[k - 1].inverse();
for (int i = k - 2; i > 0; --i)
w[i] = w[i + 1] * w[i + 1], y[i] = y[i + 1] * y[i + 1];
dw[1] = w[1], dy[1] = y[1], dw[2] = w[2], dy[2] = y[2];
for (int i = 3; i < k; ++i) {
dw[i] = dw[i - 1] * y[i - 2] * w[i];
dy[i] = dy[i - 1] * w[i - 2] * y[i];
}
}
NTT() { setwy(level); }
void fft4(vector<mint> &a, int k) {
if ((int)a.size() <= 1) return;
if (k == 1) {
mint a1 = a[1];
a[1] = a[0] - a[1];
a[0] = a[0] + a1;
return;
}
if (k & 1) {
int v = 1 << (k - 1);
for (int j = 0; j < v; ++j) {
mint ajv = a[j + v];
a[j + v] = a[j] - ajv;
a[j] += ajv;
}
}
int u = 1 << (2 + (k & 1));
int v = 1 << (k - 2 - (k & 1));
mint one = mint(1);
mint imag = dw[1];
while (v) {
// jh = 0
{
int j0 = 0;
int j1 = v;
int j2 = j1 + v;
int j3 = j2 + v;
for (; j0 < v; ++j0, ++j1, ++j2, ++j3) {
mint t0 = a[j0], t1 = a[j1], t2 = a[j2], t3 = a[j3];
mint t0p2 = t0 + t2, t1p3 = t1 + t3;
mint t0m2 = t0 - t2, t1m3 = (t1 - t3) * imag;
a[j0] = t0p2 + t1p3, a[j1] = t0p2 - t1p3;
a[j2] = t0m2 + t1m3, a[j3] = t0m2 - t1m3;
}
}
// jh >= 1
mint ww = one, xx = one * dw[2], wx = one;
for (int jh = 4; jh < u;) {
ww = xx * xx, wx = ww * xx;
int j0 = jh * v;
int je = j0 + v;
int j2 = je + v;
for (; j0 < je; ++j0, ++j2) {
mint t0 = a[j0], t1 = a[j0 + v] * xx, t2 = a[j2] * ww,
t3 = a[j2 + v] * wx;
mint t0p2 = t0 + t2, t1p3 = t1 + t3;
mint t0m2 = t0 - t2, t1m3 = (t1 - t3) * imag;
a[j0] = t0p2 + t1p3, a[j0 + v] = t0p2 - t1p3;
a[j2] = t0m2 + t1m3, a[j2 + v] = t0m2 - t1m3;
}
xx *= dw[__builtin_ctzll((jh += 4))];
}
u <<= 2;
v >>= 2;
}
}
void ifft4(vector<mint> &a, int k) {
if ((int)a.size() <= 1) return;
if (k == 1) {
mint a1 = a[1];
a[1] = a[0] - a[1];
a[0] = a[0] + a1;
return;
}
int u = 1 << (k - 2);
int v = 1;
mint one = mint(1);
mint imag = dy[1];
while (u) {
// jh = 0
{
int j0 = 0;
int j1 = v;
int j2 = v + v;
int j3 = j2 + v;
for (; j0 < v; ++j0, ++j1, ++j2, ++j3) {
mint t0 = a[j0], t1 = a[j1], t2 = a[j2], t3 = a[j3];
mint t0p1 = t0 + t1, t2p3 = t2 + t3;
mint t0m1 = t0 - t1, t2m3 = (t2 - t3) * imag;
a[j0] = t0p1 + t2p3, a[j2] = t0p1 - t2p3;
a[j1] = t0m1 + t2m3, a[j3] = t0m1 - t2m3;
}
}
// jh >= 1
mint ww = one, xx = one * dy[2], yy = one;
u <<= 2;
for (int jh = 4; jh < u;) {
ww = xx * xx, yy = xx * imag;
int j0 = jh * v;
int je = j0 + v;
int j2 = je + v;
for (; j0 < je; ++j0, ++j2) {
mint t0 = a[j0], t1 = a[j0 + v], t2 = a[j2], t3 = a[j2 + v];
mint t0p1 = t0 + t1, t2p3 = t2 + t3;
mint t0m1 = (t0 - t1) * xx, t2m3 = (t2 - t3) * yy;
a[j0] = t0p1 + t2p3, a[j2] = (t0p1 - t2p3) * ww;
a[j0 + v] = t0m1 + t2m3, a[j2 + v] = (t0m1 - t2m3) * ww;
}
xx *= dy[__builtin_ctzll(jh += 4)];
}
u >>= 4;
v <<= 2;
}
if (k & 1) {
u = 1 << (k - 1);
for (int j = 0; j < u; ++j) {
mint ajv = a[j] - a[j + u];
a[j] += a[j + u];
a[j + u] = ajv;
}
}
}
void ntt(vector<mint> &a) {
if ((int)a.size() <= 1) return;
fft4(a, __builtin_ctz(a.size()));
}
void intt(vector<mint> &a) {
if ((int)a.size() <= 1) return;
ifft4(a, __builtin_ctz(a.size()));
mint iv = mint(a.size()).inverse();
for (auto &x : a) x *= iv;
}
vector<mint> multiply(const vector<mint> &a, const vector<mint> &b) {
int l = a.size() + b.size() - 1;
if (min<int>(a.size(), b.size()) <= 40) {
vector<mint> s(l);
for (int i = 0; i < (int)a.size(); ++i)
for (int j = 0; j < (int)b.size(); ++j) s[i + j] += a[i] * b[j];
return s;
}
int k = 2, M = 4;
while (M < l) M <<= 1, ++k;
setwy(k);
vector<mint> s(M);
for (int i = 0; i < (int)a.size(); ++i) s[i] = a[i];
fft4(s, k);
if (a.size() == b.size() && a == b) {
for (int i = 0; i < M; ++i) s[i] *= s[i];
} else {
vector<mint> t(M);
for (int i = 0; i < (int)b.size(); ++i) t[i] = b[i];
fft4(t, k);
for (int i = 0; i < M; ++i) s[i] *= t[i];
}
ifft4(s, k);
s.resize(l);
mint invm = mint(M).inverse();
for (int i = 0; i < l; ++i) s[i] *= invm;
return s;
}
void ntt_doubling(vector<mint> &a) {
int M = (int)a.size();
auto b = a;
intt(b);
mint r = 1, zeta = mint(pr).pow((mint::get_mod() - 1) / (M << 1));
for (int i = 0; i < M; i++) b[i] *= r, r *= zeta;
ntt(b);
copy(begin(b), end(b), back_inserter(a));
}
};
namespace ArbitraryNTT {
using i64 = int64_t;
using u128 = __uint128_t;
constexpr int32_t m0 = 167772161;
constexpr int32_t m1 = 469762049;
constexpr int32_t m2 = 754974721;
using mint0 = LazyMontgomeryModInt<m0>;
using mint1 = LazyMontgomeryModInt<m1>;
using mint2 = LazyMontgomeryModInt<m2>;
constexpr int r01 = mint1(m0).inverse().get();
constexpr int r02 = mint2(m0).inverse().get();
constexpr int r12 = mint2(m1).inverse().get();
constexpr int r02r12 = i64(r02) * r12 % m2;
constexpr i64 w1 = m0;
constexpr i64 w2 = i64(m0) * m1;
template <typename T, typename submint>
vector<submint> mul(const vector<T> &a, const vector<T> &b) {
static NTT<submint> ntt;
vector<submint> s(a.size()), t(b.size());
for (int i = 0; i < (int)a.size(); ++i) s[i] = i64(a[i] % submint::get_mod());
for (int i = 0; i < (int)b.size(); ++i) t[i] = i64(b[i] % submint::get_mod());
return ntt.multiply(s, t);
}
template <typename T>
vector<int> multiply(const vector<T> &s, const vector<T> &t, int mod) {
auto d0 = mul<T, mint0>(s, t);
auto d1 = mul<T, mint1>(s, t);
auto d2 = mul<T, mint2>(s, t);
int n = d0.size();
vector<int> ret(n);
const int W1 = w1 % mod;
const int W2 = w2 % mod;
for (int i = 0; i < n; i++) {
int n1 = d1[i].get(), n2 = d2[i].get(), a = d0[i].get();
int b = i64(n1 + m1 - a) * r01 % m1;
int c = (i64(n2 + m2 - a) * r02r12 + i64(m2 - b) * r12) % m2;
ret[i] = (i64(a) + i64(b) * W1 + i64(c) * W2) % mod;
}
return ret;
}
template <typename mint>
vector<mint> multiply(const vector<mint> &a, const vector<mint> &b) {
if (a.size() == 0 && b.size() == 0) return {};
if (min<int>(a.size(), b.size()) < 128) {
vector<mint> ret(a.size() + b.size() - 1);
for (int i = 0; i < (int)a.size(); ++i)
for (int j = 0; j < (int)b.size(); ++j) ret[i + j] += a[i] * b[j];
return ret;
}
vector<int> s(a.size()), t(b.size());
for (int i = 0; i < (int)a.size(); ++i) s[i] = a[i].get();
for (int i = 0; i < (int)b.size(); ++i) t[i] = b[i].get();
vector<int> u = multiply<int>(s, t, mint::get_mod());
vector<mint> ret(u.size());
for (int i = 0; i < (int)u.size(); ++i) ret[i] = mint(u[i]);
return ret;
}
template <typename T>
vector<u128> multiply_u128(const vector<T> &s, const vector<T> &t) {
if (s.size() == 0 && t.size() == 0) return {};
if (min<int>(s.size(), t.size()) < 128) {
vector<u128> ret(s.size() + t.size() - 1);
for (int i = 0; i < (int)s.size(); ++i)
for (int j = 0; j < (int)t.size(); ++j) ret[i + j] += i64(s[i]) * t[j];
return ret;
}
auto d0 = mul<T, mint0>(s, t);
auto d1 = mul<T, mint1>(s, t);
auto d2 = mul<T, mint2>(s, t);
int n = d0.size();
vector<u128> ret(n);
for (int i = 0; i < n; i++) {
i64 n1 = d1[i].get(), n2 = d2[i].get();
i64 a = d0[i].get();
i64 b = (n1 + m1 - a) * r01 % m1;
i64 c = ((n2 + m2 - a) * r02r12 + (m2 - b) * r12) % m2;
ret[i] = a + b * w1 + u128(c) * w2;
}
return ret;
}
} // namespace ArbitraryNTT
namespace MultiPrecisionIntegerImpl {
struct TENS {
static constexpr int offset = 30;
constexpr TENS() : _tend() {
_tend[offset] = 1;
for (int i = 1; i <= offset; i++) {
_tend[offset + i] = _tend[offset + i - 1] * 10.0;
_tend[offset - i] = 1.0 / _tend[offset + i];
}
}
long double ten_ld(int n) const {
assert(-offset <= n and n <= offset);
return _tend[n + offset];
}
private:
long double _tend[offset * 2 + 1];
};
} // namespace MultiPrecisionIntegerImpl
// 0 は neg=false, dat={} として扱う
struct MultiPrecisionInteger {
using M = MultiPrecisionInteger;
inline constexpr static MultiPrecisionIntegerImpl::TENS tens = {};
static constexpr int D = 1000000000;
static constexpr int logD = 9;
bool neg;
vector<int> dat;
MultiPrecisionInteger() : neg(false), dat() {}
MultiPrecisionInteger(bool n, const vector<int>& d) : neg(n), dat(d) {}
template <typename I,
enable_if_t<internal::is_broadly_integral_v<I>>* = nullptr>
MultiPrecisionInteger(I x) : neg(false) {
if constexpr (internal::is_broadly_signed_v<I>) {
if (x < 0) neg = true, x = -x;
}
while (x) dat.push_back(x % D), x /= D;
}
MultiPrecisionInteger(const string& S) : neg(false) {
assert(!S.empty());
if (S.size() == 1u && S[0] == '0') return;
int l = 0;
if (S[0] == '-') ++l, neg = true;
for (int ie = S.size(); l < ie; ie -= logD) {
int is = max(l, ie - logD);
long long x = 0;
for (int i = is; i < ie; i++) x = x * 10 + S[i] - '0';
dat.push_back(x);
}
}
friend M operator+(const M& lhs, const M& rhs) {
if (lhs.neg == rhs.neg) return {lhs.neg, _add(lhs.dat, rhs.dat)};
if (_leq(lhs.dat, rhs.dat)) {
// |l| <= |r|
auto c = _sub(rhs.dat, lhs.dat);
bool n = _is_zero(c) ? false : rhs.neg;
return {n, c};
}
auto c = _sub(lhs.dat, rhs.dat);
bool n = _is_zero(c) ? false : lhs.neg;
return {n, c};
}
friend M operator-(const M& lhs, const M& rhs) { return lhs + (-rhs); }
friend M operator*(const M& lhs, const M& rhs) {
auto c = _mul(lhs.dat, rhs.dat);
bool n = _is_zero(c) ? false : (lhs.neg ^ rhs.neg);
return {n, c};
}
friend pair<M, M> divmod(const M& lhs, const M& rhs) {
auto dm = _divmod_newton(lhs.dat, rhs.dat);
bool dn = _is_zero(dm.first) ? false : lhs.neg != rhs.neg;
bool mn = _is_zero(dm.second) ? false : lhs.neg;
return {M{dn, dm.first}, M{mn, dm.second}};
}
friend M operator/(const M& lhs, const M& rhs) {
return divmod(lhs, rhs).first;
}
friend M operator%(const M& lhs, const M& rhs) {
return divmod(lhs, rhs).second;
}
M& operator+=(const M& rhs) { return (*this) = (*this) + rhs; }
M& operator-=(const M& rhs) { return (*this) = (*this) - rhs; }
M& operator*=(const M& rhs) { return (*this) = (*this) * rhs; }
M& operator/=(const M& rhs) { return (*this) = (*this) / rhs; }
M& operator%=(const M& rhs) { return (*this) = (*this) % rhs; }
M operator-() const {
if (is_zero()) return *this;
return {!neg, dat};
}
M operator+() const { return *this; }
friend M abs(const M& m) { return {false, m.dat}; }
bool is_zero() const { return _is_zero(dat); }
friend bool operator==(const M& lhs, const M& rhs) {
return lhs.neg == rhs.neg && lhs.dat == rhs.dat;
}
friend bool operator!=(const M& lhs, const M& rhs) {
return lhs.neg != rhs.neg || lhs.dat != rhs.dat;
}
friend bool operator<(const M& lhs, const M& rhs) {
if (lhs == rhs) return false;
return _neq_lt(lhs, rhs);
}
friend bool operator<=(const M& lhs, const M& rhs) {
if (lhs == rhs) return true;
return _neq_lt(lhs, rhs);
}
friend bool operator>(const M& lhs, const M& rhs) {
if (lhs == rhs) return false;
return _neq_lt(rhs, lhs);
}
friend bool operator>=(const M& lhs, const M& rhs) {
if (lhs == rhs) return true;
return _neq_lt(rhs, lhs);
}
// a * 10^b (1 <= |a| < 10) の形で渡す
// 相対誤差：10^{-16} ~ 10^{-19} 程度 (処理系依存)
pair<long double, int> dfp() const {
if (is_zero()) return {0, 0};
int l = max<int>(0, _size() - 3);
int b = logD * l;
string prefix{};
for (int i = _size() - 1; i >= l; i--) {
prefix += _itos(dat[i], i != _size() - 1);
}
b += prefix.size() - 1;
long double a = 0;
for (auto& c : prefix) a = a * 10.0 + (c - '0');
a *= tens.ten_ld(-((int)prefix.size()) + 1);
a = clamp<long double>(a, 1.0, nextafterl(10.0, 1.0));
if (neg) a = -a;
return {a, b};
}
string to_string() const {
if (is_zero()) return "0";
string res;
if (neg) res.push_back('-');
for (int i = _size() - 1; i >= 0; i--) {
res += _itos(dat[i], i != _size() - 1);
}
return res;
}
long double to_ld() const {
auto [a, b] = dfp();
if (-tens.offset <= b and b <= tens.offset) {
return a * tens.ten_ld(b);
}
return a * powl(10, b);
}
long long to_ll() const {
long long res = _to_ll(dat);
return neg ? -res : res;
}
__int128_t to_i128() const {
__int128_t res = _to_i128(dat);
return neg ? -res : res;
}
friend istream& operator>>(istream& is, M& m) {
string s;
is >> s;
m = M{s};
return is;
}
friend ostream& operator<<(ostream& os, const M& m) {
return os << m.to_string();
}
// 内部の関数をテスト
static void _test_private_function(const M&, const M&);
private:
// size
int _size() const { return dat.size(); }
// a == b
static bool _eq(const vector<int>& a, const vector<int>& b) { return a == b; }
// a < b
static bool _lt(const vector<int>& a, const vector<int>& b) {
if (a.size() != b.size()) return a.size() < b.size();
for (int i = a.size() - 1; i >= 0; i--) {
if (a[i] != b[i]) return a[i] < b[i];
}
return false;
}
// a <= b
static bool _leq(const vector<int>& a, const vector<int>& b) {
return _eq(a, b) || _lt(a, b);
}
// a < b (s.t. a != b)
static bool _neq_lt(const M& lhs, const M& rhs) {
assert(lhs != rhs);
if (lhs.neg != rhs.neg) return lhs.neg;
bool f = _lt(lhs.dat, rhs.dat);
if (f) return !lhs.neg;
return lhs.neg;
}
// a == 0
static bool _is_zero(const vector<int>& a) { return a.empty(); }
// a == 1
static bool _is_one(const vector<int>& a) {
return (int)a.size() == 1 && a[0] == 1;
}
// 末尾 0 を削除
static void _shrink(vector<int>& a) {
while (a.size() && a.back() == 0) a.pop_back();
}
// 末尾 0 を削除
void _shrink() {
while (_size() && dat.back() == 0) dat.pop_back();
}
// a + b
static vector<int> _add(const vector<int>& a, const vector<int>& b) {
vector<int> c(max(a.size(), b.size()) + 1);
for (int i = 0; i < (int)a.size(); i++) c[i] += a[i];
for (int i = 0; i < (int)b.size(); i++) c[i] += b[i];
for (int i = 0; i < (int)c.size() - 1; i++) {
if (c[i] >= D) c[i] -= D, c[i + 1]++;
}
_shrink(c);
return c;
}
// a - b
static vector<int> _sub(const vector<int>& a, const vector<int>& b) {
assert(_leq(b, a));
vector<int> c{a};
int borrow = 0;
for (int i = 0; i < (int)a.size(); i++) {
if (i < (int)b.size()) borrow += b[i];
c[i] -= borrow;
borrow = 0;
if (c[i] < 0) c[i] += D, borrow = 1;
}
assert(borrow == 0);
_shrink(c);
return c;
}
// a * b (fft)
static vector<int> _mul_fft(const vector<int>& a, const vector<int>& b) {
if (a.empty() || b.empty()) return {};
auto m = ArbitraryNTT::multiply_u128(a, b);
vector<int> c;
c.reserve(m.size() + 3);
__uint128_t x = 0;
for (int i = 0;; i++) {
if (i >= (int)m.size() && x == 0) break;
if (i < (int)m.size()) x += m[i];
c.push_back(x % D);
x /= D;
}
_shrink(c);
return c;
}
// a * b (naive)
static vector<int> _mul_naive(const vector<int>& a, const vector<int>& b) {
if (a.empty() || b.empty()) return {};
vector<long long> prod(a.size() + b.size() - 1 + 1);
for (int i = 0; i < (int)a.size(); i++) {
for (int j = 0; j < (int)b.size(); j++) {
long long p = 1LL * a[i] * b[j];
prod[i + j] += p;
if (prod[i + j] >= (4LL * D * D)) {
prod[i + j] -= 4LL * D * D;
prod[i + j + 1] += 4LL * D;
}
}
}
vector<int> c(prod.size() + 1);
long long x = 0;
int i = 0;
for (; i < (int)prod.size(); i++) x += prod[i], c[i] = x % D, x /= D;
while (x) c[i] = x % D, x /= D, i++;
_shrink(c);
return c;
}
// a * b
static vector<int> _mul(const vector<int>& a, const vector<int>& b) {
if (_is_zero(a) || _is_zero(b)) return {};
if (_is_one(a)) return b;
if (_is_one(b)) return a;
if (min<int>(a.size(), b.size()) <= 128) {
return a.size() < b.size() ? _mul_naive(b, a) : _mul_naive(a, b);
}
return _mul_fft(a, b);
}
// 0 <= A < 1e18, 1 <= B < 1e9
static pair<vector<int>, vector<int>> _divmod_li(const vector<int>& a,
const vector<int>& b) {
assert(0 <= (int)a.size() && (int)a.size() <= 2);
assert((int)b.size() == 1);
long long va = _to_ll(a);
int vb = b[0];
return {_integer_to_vec(va / vb), _integer_to_vec(va % vb)};
}
// 0 <= A < 1e18, 1 <= B < 1e18
static pair<vector<int>, vector<int>> _divmod_ll(const vector<int>& a,
const vector<int>& b) {
assert(0 <= (int)a.size() && (int)a.size() <= 2);
assert(1 <= (int)b.size() && (int)b.size() <= 2);
long long va = _to_ll(a), vb = _to_ll(b);
return {_integer_to_vec(va / vb), _integer_to_vec(va % vb)};
}
// 1 <= B < 1e9
static pair<vector<int>, vector<int>> _divmod_1e9(const vector<int>& a,
const vector<int>& b) {
assert((int)b.size() == 1);
if (b[0] == 1) return {a, {}};
if ((int)a.size() <= 2) return _divmod_li(a, b);
vector<int> quo(a.size());
long long d = 0;
int b0 = b[0];
for (int i = a.size() - 1; i >= 0; i--) {
d = d * D + a[i];
assert(d < 1LL * D * b0);
int q = d / b0, r = d % b0;
quo[i] = q, d = r;
}
_shrink(quo);
return {quo, d ? vector<int>{int(d)} : vector<int>{}};
}
// 0 <= A, 1 <= B
static pair<vector<int>, vector<int>> _divmod_naive(const vector<int>& a,
const vector<int>& b) {
if (_is_zero(b)) {
cerr << "Divide by Zero Exception" << endl;
exit(1);
}
assert(1 <= (int)b.size());
if ((int)b.size() == 1) return _divmod_1e9(a, b);
if (max<int>(a.size(), b.size()) <= 2) return _divmod_ll(a, b);
if (_lt(a, b)) return {{}, a};
// B >= 1e9, A >= B
int norm = D / (b.back() + 1);
vector<int> x = _mul(a, {norm});
vector<int> y = _mul(b, {norm});
int yb = y.back();
vector<int> quo(x.size() - y.size() + 1);
vector<int> rem(x.end() - y.size(), x.end());
for (int i = quo.size() - 1; i >= 0; i--) {
if (rem.size() < y.size()) {
// do nothing
} else if (rem.size() == y.size()) {
if (_leq(y, rem)) {
quo[i] = 1, rem = _sub(rem, y);
}
} else {
assert(y.size() + 1 == rem.size());
long long rb = 1LL * rem[rem.size() - 1] * D + rem[rem.size() - 2];
int q = rb / yb;
vector<int> yq = _mul(y, {q});
// 真の商は q-2 以上 q+1 以下だが自信が無いので念のため while を回す
while (_lt(rem, yq)) q--, yq = _sub(yq, y);
rem = _sub(rem, yq);
while (_leq(y, rem)) q++, rem = _sub(rem, y);
quo[i] = q;
}
if (i) rem.insert(begin(rem), x[i - 1]);
}
_shrink(quo), _shrink(rem);
auto [q2, r2] = _divmod_1e9(rem, {norm});
assert(_is_zero(r2));
return {quo, q2};
}
// 0 <= A, 1 <= B
static pair<vector<int>, vector<int>> _divmod_dc(const vector<int>& a,
const vector<int>& b);
// 1 / a を 絶対誤差 B^{-deg} で求める
static vector<int> _calc_inv(const vector<int>& a, int deg) {
assert(!a.empty() && D / 2 <= a.back() and a.back() < D);
int k = deg, c = a.size();
while (k > 64) k = (k + 1) / 2;
vector<int> z(c + k + 1);
z.back() = 1;
z = _divmod_naive(z, a).first;
while (k < deg) {
vector<int> s = _mul(z, z);
s.insert(begin(s), 0);
int d = min(c, 2 * k + 1);
vector<int> t{end(a) - d, end(a)}, u = _mul(s, t);
u.erase(begin(u), begin(u) + d);
vector<int> w(k + 1), w2 = _add(z, z);
copy(begin(w2), end(w2), back_inserter(w));
z = _sub(w, u);
z.erase(begin(z));
k *= 2;
}
z.erase(begin(z), begin(z) + k - deg);
return z;
}
static pair<vector<int>, vector<int>> _divmod_newton(const vector<int>& a,
const vector<int>& b) {
if (_is_zero(b)) {
cerr << "Divide by Zero Exception" << endl;
exit(1);
}
if ((int)b.size() <= 64) return _divmod_naive(a, b);
if ((int)a.size() - (int)b.size() <= 64) return _divmod_naive(a, b);
int norm = D / (b.back() + 1);
vector<int> x = _mul(a, {norm});
vector<int> y = _mul(b, {norm});
int s = x.size(), t = y.size();
int deg = s - t + 2;
vector<int> z = _calc_inv(y, deg);
vector<int> q = _mul(x, z);
q.erase(begin(q), begin(q) + t + deg);
vector<int> yq = _mul(y, {q});
while (_lt(x, yq)) q = _sub(q, {1}), yq = _sub(yq, y);
vector<int> r = _sub(x, yq);
while (_leq(y, r)) q = _add(q, {1}), r = _sub(r, y);
_shrink(q), _shrink(r);
auto [q2, r2] = _divmod_1e9(r, {norm});
assert(_is_zero(r2));
return {q, q2};
}
// int -> string
// 先頭かどうかに応じて zero padding するかを決める
static string _itos(int x, bool zero_padding) {
assert(0 <= x && x < D);
string res;
for (int i = 0; i < logD; i++) {
res.push_back('0' + x % 10), x /= 10;
}
if (!zero_padding) {
while (res.size() && res.back() == '0') res.pop_back();
assert(!res.empty());
}
reverse(begin(res), end(res));
return res;
}
// convert ll to vec
template <typename I,
enable_if_t<internal::is_broadly_integral_v<I>>* = nullptr>
static vector<int> _integer_to_vec(I x) {
if constexpr (internal::is_broadly_signed_v<I>) {
assert(x >= 0);
}
vector<int> res;
while (x) res.push_back(x % D), x /= D;
return res;
}
static long long _to_ll(const vector<int>& a) {
long long res = 0;
for (int i = (int)a.size() - 1; i >= 0; i--) res = res * D + a[i];
return res;
}
static __int128_t _to_i128(const vector<int>& a) {
__int128_t res = 0;
for (int i = (int)a.size() - 1; i >= 0; i--) res = res * D + a[i];
return res;
}
static void _dump(const vector<int>& a, string s = "") {
if (!s.empty()) cerr << s << " : ";
cerr << "{ ";
for (int i = 0; i < (int)a.size(); i++) cerr << a[i] << ", ";
cerr << "}" << endl;
}
};
using bigint = MultiPrecisionInteger;
/**
* @brief 多倍長整数
*/
namespace GCDforBigintImpl {
bigint bigint_pow(bigint a, long long k) {
bigint res{1};
while (k) {
if (k & 1) res = res * a;
k >>= 1;
if (k) a = a * a;
}
return res;
}
// a = 2^x 5^y の形で表現する
pair<int, int> shrink(int a) {
assert(a > 0);
int x = __builtin_ctz(a);
a >>= x;
int y = a == 1 ? 0
: a == 5 ? 1
: a == 25 ? 2
: a == 125 ? 3
: a == 625 ? 4
: a == 3125 ? 5
: a == 15625 ? 6
: a == 78125 ? 7
: a == 390625 ? 8
: 9;
return {x, y};
}
pair<int, int> shrink(bigint& a) {
assert(a.neg == false);
if (a.dat.empty()) return {0, 0};
pair<int, int> res{0, 0};
while (true) {
int g = gcd(bigint::D, a.dat[0]);
if (g == 1) break;
if (g != bigint::D) a *= bigint::D / g;
a.dat.erase(begin(a.dat));
auto s = shrink(g);
res.first += s.first, res.second += s.second;
}
return res;
}
template <bool FAST = false>
bigint gcd_d_ary(bigint a, bigint b) {
a.neg = b.neg = false;
if constexpr (FAST) {
if (max<int>(a.dat.size(), b.dat.size()) <= 4) {
return __int128_t(BinaryGCDImpl::binary_gcd128(a.to_i128(), b.to_i128()));
}
}
if (a.dat.empty()) return b;
if (b.dat.empty()) return a;
pair<int, int> s = shrink(a), t = shrink(b);
if (a < b) swap(a, b);
while (true) {
if (b.dat.empty()) break;
if constexpr (FAST) {
if ((int)a.dat.size() <= 4) break;
}
a = a - b;
if (!a.dat.empty()) {
while (true) {
int g = gcd<int>(a.dat[0], bigint::D);
if (g == 1) break;
if (g != bigint::D) a *= bigint::D / g;
a.dat.erase(begin(a.dat));
}
}
if (a < b) swap(a, b);
}
assert(a >= b);
bigint g;
if constexpr (FAST) {
if (b.dat.empty()) {
g = a;
} else {
g = __int128_t(BinaryGCDImpl::binary_gcd128(a.to_i128(), b.to_i128()));
}
} else {
g = a;
}
int e2 = min(s.first, t.first);
int e5 = min(s.second, t.second);
if (e2) g *= bigint_pow(bigint{2}, e2);
if (e5) g *= bigint_pow(bigint{5}, e5);
return g;
}
} // namespace GCDforBigintImpl
MultiPrecisionInteger gcd(const MultiPrecisionInteger& a,
const MultiPrecisionInteger& b) {
return GCDforBigintImpl::gcd_d_ary<true>(a, b);
}
using BigRational = RationalBase<bigint, bigint>;
double to_double(const BigRational& r) {
pair<long double, int> a = r.x.dfp();
pair<long double, int> b = r.y.dfp();
return a.first / b.first * powl(10.0, a.second - b.second);
}
//
using mint = BigRational;
using vm = vector<mint>;
using vvm = vector<vm>;
using fps = FormalPowerSeries_rational<mint>;
Binomial_rational<mint> C;
// enumerate x : x \subset b
vector<int> enumerate_subset(int b) {
vector<int> res;
for (int i = b; i >= 0; --i) res.push_back(i &= b);
return res;
};
// enumerate x : x \in {n} and x \superset b
vector<int> enumerate_superset(int b, int n) {
vector<int> res;
for (int i = b; i < (1 << n); i = (i + 1) | b) res.push_back(i);
return res;
}
/**
* @brief 下位集合/上位集合の列挙
*/
using namespace Nyaan;
using R = BigRational;
void q() {
inl(N);
vl a(N);
in(a);
VV<Rational> dp(PW(N));
rep(i, N) dp[PW(i)].push_back(a[i]);
rep(i, PW(N)) {
if (popcnt(i) <= 1) continue;
each(j, enumerate_subset(i)) {
if (j < i - j) continue;
each(x, dp[i - j]) {
each(y, dp[j]) {
if (x == y) die("YES");
if (i != MSK(N)) {
dp[i].push_back(x + y);
dp[i].push_back(x - y);
dp[i].push_back(x * y);
dp[i].push_back(x / y);
}
}
}
}
dp[i] = mkuni(dp[i]);
}
out("NO");
}
void Nyaan::solve() {
int t = 1;
// in(t);
while (t--) q();
}