結果
| 問題 | No.3415 Dial Lock |
| コンテスト | |
| ユーザー |
 KumaTachiRen
|
| 提出日時 | 2025-12-22 01:11:12 |
| 言語 | C++23 (gcc 13.3.0 + boost 1.89.0) |
| 結果 |
AC
|
| 実行時間 | 2,823 ms / 10,000 ms |
| コード長 | 28,700 bytes |
| 記録 | |
| コンパイル時間 | 4,312 ms |
| コンパイル使用メモリ | 318,784 KB |
| 実行使用メモリ | 41,152 KB |
| 最終ジャッジ日時 | 2025-12-22 01:12:03 |
| 合計ジャッジ時間 | 46,716 ms |
|
ジャッジサーバーID （参考情報） |
judge5 / judge4 |
（要ログイン）
| ファイルパターン | 結果 |
|---|---|
| sample | AC * 3 |
| other | AC * 18 |
ソースコード
#line 2 "library/template/template.hpp"
#include <bits/stdc++.h>
using namespace std;
#line 2 "library/template/macro.hpp"
#define rep(i, a, b) for (int i = (a); i < (int)(b); i++)
#define rrep(i, a, b) for (int i = (int)(b) - 1; i >= (a); i--)
#define ALL(v) (v).begin(), (v).end()
#define UNIQUE(v) sort(ALL(v)), (v).erase(unique(ALL(v)), (v).end())
#define SZ(v) (int)v.size()
#define MIN(v) *min_element(ALL(v))
#define MAX(v) *max_element(ALL(v))
#define LB(v, x) int(lower_bound(ALL(v), (x)) - (v).begin())
#define UB(v, x) int(upper_bound(ALL(v), (x)) - (v).begin())
#define YN(b) cout << ((b) ? "YES" : "NO") << "\n";
#define Yn(b) cout << ((b) ? "Yes" : "No") << "\n";
#define yn(b) cout << ((b) ? "yes" : "no") << "\n";
#line 6 "library/template/template.hpp"
#line 2 "library/template/util.hpp"
using uint = unsigned int;
using ll = long long int;
using ull = unsigned long long;
using i128 = __int128_t;
using u128 = __uint128_t;
template <class T, class S = T>
S SUM(const vector<T>& a) {
return accumulate(ALL(a), S(0));
}
template <class T>
inline bool chmin(T& a, T b) {
if (a > b) {
a = b;
return true;
}
return false;
}
template <class T>
inline bool chmax(T& a, T b) {
if (a < b) {
a = b;
return true;
}
return false;
}
template <class T>
int popcnt(T x) {
return __builtin_popcountll(x);
}
template <class T>
int topbit(T x) {
return (x == 0 ? -1 : 63 - __builtin_clzll(x));
}
template <class T>
int lowbit(T x) {
return (x == 0 ? -1 : __builtin_ctzll(x));
}
#line 8 "library/template/template.hpp"
#line 2 "library/template/inout.hpp"
struct Fast {
Fast() {
cin.tie(nullptr);
ios_base::sync_with_stdio(false);
cout << fixed << setprecision(15);
}
} fast;
template <class T1, class T2>
istream& operator>>(istream& is, pair<T1, T2>& p) {
return is >> p.first >> p.second;
}
template <class T1, class T2>
ostream& operator<<(ostream& os, const pair<T1, T2>& p) {
return os << p.first << " " << p.second;
}
template <class T>
istream& operator>>(istream& is, vector<T>& a) {
for (auto& v : a) is >> v;
return is;
}
template <class T>
ostream& operator<<(ostream& os, const vector<T>& a) {
for (auto it = a.begin(); it != a.end();) {
os << *it;
if (++it != a.end()) os << " ";
}
return os;
}
template <class T>
ostream& operator<<(ostream& os, const set<T>& st) {
os << "{";
for (auto it = st.begin(); it != st.end();) {
os << *it;
if (++it != st.end()) os << ",";
}
os << "}";
return os;
}
template <class T1, class T2>
ostream& operator<<(ostream& os, const map<T1, T2>& mp) {
os << "{";
for (auto it = mp.begin(); it != mp.end();) {
os << it->first << ":" << it->second;
if (++it != mp.end()) os << ",";
}
os << "}";
return os;
}
ostream& operator<<(ostream& os, __uint128_t x) {
char buf[40];
size_t k = 0;
while (x > 0) buf[k++] = (char)(x % 10 + '0'), x /= 10;
if (k == 0) buf[k++] = '0';
while (k) cout << buf[--k];
return cout;
}
ostream& operator<<(ostream& os, __int128_t x) {
return x < 0 ? (cout << '-' << (__uint128_t)(-x)) : (cout << (__uint128_t)x);
}
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...);
}
#line 10 "library/template/template.hpp"
#line 2 "library/template/debug.hpp"
#ifdef LOCAL
#define debug 1
#define show(...) _show(0, #__VA_ARGS__, __VA_ARGS__)
#else
#define debug 0
#define show(...) true
#endif
template <class T>
void _show(int i, T name) {
cerr << '\n';
}
template <class T1, class T2, class... T3>
void _show(int i, const T1& a, const T2& b, const T3&... c) {
for (; a[i] != ',' && a[i] != '\0'; i++) cerr << a[i];
cerr << ":" << b << " ";
_show(i + 1, a, c...);
}
#line 2 "main.cpp"
#line 2 "library/math/util.hpp"
namespace Math {
template <class T>
T safe_mod(T a, T b) {
assert(b != 0);
if (b < 0) a = -a, b = -b;
a %= b;
return a >= 0 ? a : a + b;
}
template <class T>
T floor(T a, T b) {
assert(b != 0);
if (b < 0) a = -a, b = -b;
return a >= 0 ? a / b : (a + 1) / b - 1;
}
template <class T>
T ceil(T a, T b) {
assert(b != 0);
if (b < 0) a = -a, b = -b;
return a > 0 ? (a - 1) / b + 1 : a / b;
}
long long isqrt(long long n) {
if (n <= 0) return 0;
long long x = sqrt(n);
while ((x + 1) * (x + 1) <= n) x++;
while (x * x > n) x--;
return x;
}
// return g=gcd(a,b)
// a*x+b*y=g
// - b!=0 -> 0<=x<|b|/g
// - b=0 -> ax=g
template <class T>
T ext_gcd(T a, T b, T& x, T& y) {
T a0 = a, b0 = b;
bool sgn_a = a < 0, sgn_b = b < 0;
if (sgn_a) a = -a;
if (sgn_b) b = -b;
if (b == 0) {
x = sgn_a ? -1 : 1;
y = 0;
return a;
}
T x00 = 1, x01 = 0, x10 = 0, x11 = 1;
while (b != 0) {
T q = a / b, r = a - b * q;
x00 -= q * x01;
x10 -= q * x11;
swap(x00, x01);
swap(x10, x11);
a = b, b = r;
}
x = x00, y = x10;
if (sgn_a) x = -x;
if (sgn_b) y = -y;
if (b0 != 0) {
a0 /= a, b0 /= a;
if (b0 < 0) a0 = -a0, b0 = -b0;
T q = x >= 0 ? x / b0 : (x + 1) / b0 - 1;
x -= b0 * q;
y += a0 * q;
}
return a;
}
constexpr long long inv_mod(long long x, long long m) {
x %= m;
if (x < 0) x += m;
long long a = m, b = x;
long long y0 = 0, y1 = 1;
while (b > 0) {
long long q = a / b;
swap(a -= q * b, b);
swap(y0 -= q * y1, y1);
}
if (y0 < 0) y0 += m / a;
return y0;
}
long long pow_mod(long long x, long long n, long long m) {
x = (x % m + m) % m;
long long y = 1;
while (n) {
if (n & 1) y = y * x % m;
x = x * x % m;
n >>= 1;
}
return y;
}
constexpr long long pow_mod_constexpr(long long x, long long n, int m) {
if (m == 1) return 0;
unsigned int _m = (unsigned int)(m);
unsigned long long r = 1;
unsigned long long y = x % m;
if (y >= m) y += m;
while (n) {
if (n & 1) r = (r * y) % _m;
y = (y * y) % _m;
n >>= 1;
}
return r;
}
constexpr bool is_prime_constexpr(int n) {
if (n <= 1) return false;
if (n == 2 || n == 7 || n == 61) return true;
if (n % 2 == 0) return false;
long long d = n - 1;
while (d % 2 == 0) d /= 2;
constexpr long long bases[3] = {2, 7, 61};
for (long long a : bases) {
long long t = d;
long long y = pow_mod_constexpr(a, t, n);
while (t != n - 1 && y != 1 && y != n - 1) {
y = y * y % n;
t <<= 1;
}
if (y != n - 1 && t % 2 == 0) {
return false;
}
}
return true;
}
template <int n>
constexpr bool is_prime = is_prime_constexpr(n);
}; // namespace Math
#line 3 "library/modint/modint.hpp"
template <unsigned int m = 998244353>
struct ModInt {
using mint = ModInt;
static constexpr unsigned int get_mod() { return m; }
static mint raw(int v) {
mint x;
x._v = v;
return x;
}
ModInt() : _v(0) {}
ModInt(int64_t v) {
long long x = (long long)(v % (long long)(umod()));
if (x < 0) x += umod();
_v = (unsigned int)(x);
}
unsigned int val() const { return _v; }
mint& operator++() {
_v++;
if (_v == umod()) _v = 0;
return *this;
}
mint& operator--() {
if (_v == 0) _v = umod();
_v--;
return *this;
}
mint operator++(int) {
mint result = *this;
++*this;
return result;
}
mint operator--(int) {
mint result = *this;
--*this;
return result;
}
mint& operator+=(const mint& rhs) {
_v += rhs._v;
if (_v >= umod()) _v -= umod();
return *this;
}
mint& operator-=(const mint& rhs) {
_v -= rhs._v;
if (_v >= umod()) _v += umod();
return *this;
}
mint& operator*=(const mint& rhs) {
unsigned long long z = _v;
z *= rhs._v;
_v = (unsigned int)(z % umod());
return *this;
}
mint& operator/=(const mint& rhs) { return *this *= rhs.inv(); }
mint operator+() const { return *this; }
mint operator-() const { return mint() - *this; }
mint pow(long long n) const {
assert(0 <= n);
mint x = *this, r = 1;
while (n) {
if (n & 1) r *= x;
x *= x;
n >>= 1;
}
return r;
}
mint inv() const {
if (is_prime) {
assert(_v);
return pow(umod() - 2);
} else {
auto inv = Math::inv_mod(_v, umod());
return raw(inv);
}
}
friend mint operator+(const mint& lhs, const mint& rhs) { return mint(lhs) += rhs; }
friend mint operator-(const mint& lhs, const mint& rhs) { return mint(lhs) -= rhs; }
friend mint operator*(const mint& lhs, const mint& rhs) { return mint(lhs) *= rhs; }
friend mint operator/(const mint& lhs, const mint& rhs) { return mint(lhs) /= rhs; }
friend bool operator==(const mint& lhs, const mint& rhs) { return lhs._v == rhs._v; }
friend bool operator!=(const mint& lhs, const mint& rhs) { return lhs._v != rhs._v; }
friend istream& operator>>(istream& is, mint& x) {
int64_t v;
is >> v;
x = mint(v);
return is;
}
friend ostream& operator<<(ostream& os, const mint& x) { return os << x.val(); }
private:
unsigned int _v;
static constexpr unsigned int umod() { return m; }
static constexpr bool is_prime = Math::is_prime<m>;
};
#line 4 "main.cpp"
using mint = ModInt<120586241>;
#line 2 "library/fps/fps-ntt-friendly.hpp"
#line 2 "library/fft/ntt.hpp"
template <class mint>
struct NTT {
static constexpr unsigned int mod = mint::get_mod();
static constexpr unsigned long long pow_constexpr(unsigned long long x, unsigned long long n, unsigned long long m) {
unsigned long long y = 1;
while (n) {
if (n & 1) y = y * x % m;
x = x * x % m;
n >>= 1;
}
return y;
}
static constexpr unsigned int get_g() {
unsigned long long x = 2;
while (pow_constexpr(x, (mod - 1) >> 1, mod) == 1) x += 1;
return x;
}
static constexpr unsigned int g = get_g();
static constexpr int rank2 = __builtin_ctzll(mod - 1);
array<mint, rank2 + 1> root;
array<mint, rank2 + 1> iroot;
array<mint, max(0, rank2 - 2 + 1)> rate2;
array<mint, max(0, rank2 - 2 + 1)> irate2;
array<mint, max(0, rank2 - 3 + 1)> rate3;
array<mint, max(0, rank2 - 3 + 1)> irate3;
NTT() {
root[rank2] = mint(g).pow((mod - 1) >> rank2);
iroot[rank2] = root[rank2].inv();
for (int i = rank2 - 1; i >= 0; i--) {
root[i] = root[i + 1] * root[i + 1];
iroot[i] = iroot[i + 1] * iroot[i + 1];
}
{
mint prod = 1, iprod = 1;
for (int i = 0; i <= rank2 - 2; i++) {
rate2[i] = root[i + 2] * prod;
irate2[i] = iroot[i + 2] * iprod;
prod *= iroot[i + 2];
iprod *= root[i + 2];
}
}
{
mint prod = 1, iprod = 1;
for (int i = 0; i <= rank2 - 3; i++) {
rate3[i] = root[i + 3] * prod;
irate3[i] = iroot[i + 3] * iprod;
prod *= iroot[i + 3];
iprod *= root[i + 3];
}
}
}
void ntt(vector<mint>& a) {
int n = int(a.size());
int h = __builtin_ctzll((unsigned int)n);
a.resize(1 << h);
int len = 0; // a[i, i+(n>>len), i+2*(n>>len), ..] is transformed
while (len < h) {
if (h - len == 1) {
int p = 1 << (h - len - 1);
mint rot = 1;
for (int s = 0; s < (1 << len); s++) {
int offset = s << (h - len);
for (int i = 0; i < p; i++) {
auto l = a[i + offset];
auto r = a[i + offset + p] * rot;
a[i + offset] = l + r;
a[i + offset + p] = l - r;
}
if (s + 1 != (1 << len)) rot *= rate2[__builtin_ctzll(~(unsigned int)(s))];
}
len++;
} else {
// 4-base
int p = 1 << (h - len - 2);
mint rot = 1, imag = root[2];
for (int s = 0; s < (1 << len); s++) {
mint rot2 = rot * rot;
mint rot3 = rot2 * rot;
int offset = s << (h - len);
for (int i = 0; i < p; i++) {
auto mod2 = 1ULL * mint::get_mod() * mint::get_mod();
auto a0 = 1ULL * a[i + offset].val();
auto a1 = 1ULL * a[i + offset + p].val() * rot.val();
auto a2 = 1ULL * a[i + offset + 2 * p].val() * rot2.val();
auto a3 = 1ULL * a[i + offset + 3 * p].val() * rot3.val();
auto a1na3imag = 1ULL * mint(a1 + mod2 - a3).val() * imag.val();
auto na2 = mod2 - a2;
a[i + offset] = a0 + a2 + a1 + a3;
a[i + offset + 1 * p] = a0 + a2 + (2 * mod2 - (a1 + a3));
a[i + offset + 2 * p] = a0 + na2 + a1na3imag;
a[i + offset + 3 * p] = a0 + na2 + (mod2 - a1na3imag);
}
if (s + 1 != (1 << len)) rot *= rate3[__builtin_ctzll(~(unsigned int)(s))];
}
len += 2;
}
}
}
void intt(vector<mint>& a) {
int n = int(a.size());
int h = __builtin_ctzll((unsigned int)n);
a.resize(1 << h);
int len = h; // a[i, i+(n>>len), i+2*(n>>len), ..] is transformed
while (len) {
if (len == 1) {
int p = 1 << (h - len);
mint irot = 1;
for (int s = 0; s < (1 << (len - 1)); s++) {
int offset = s << (h - len + 1);
for (int i = 0; i < p; i++) {
auto l = a[i + offset];
auto r = a[i + offset + p];
a[i + offset] = l + r;
a[i + offset + p] = (unsigned long long)(mint::get_mod() + l.val() - r.val()) * irot.val();
}
if (s + 1 != (1 << (len - 1))) irot *= irate2[__builtin_ctzll(~(unsigned int)(s))];
}
len--;
} else {
// 4-base
int p = 1 << (h - len);
mint irot = 1, iimag = iroot[2];
for (int s = 0; s < (1 << (len - 2)); s++) {
mint irot2 = irot * irot;
mint irot3 = irot2 * irot;
int offset = s << (h - len + 2);
for (int i = 0; i < p; i++) {
auto a0 = 1ULL * a[i + offset + 0 * p].val();
auto a1 = 1ULL * a[i + offset + 1 * p].val();
auto a2 = 1ULL * a[i + offset + 2 * p].val();
auto a3 = 1ULL * a[i + offset + 3 * p].val();
auto a2na3iimag = 1ULL * mint((mint::get_mod() + a2 - a3) * iimag.val()).val();
a[i + offset] = a0 + a1 + a2 + a3;
a[i + offset + 1 * p] = (a0 + (mint::get_mod() - a1) + a2na3iimag) * irot.val();
a[i + offset + 2 * p] = (a0 + a1 + (mint::get_mod() - a2) + (mint::get_mod() - a3)) * irot2.val();
a[i + offset + 3 * p] = (a0 + (mint::get_mod() - a1) + (mint::get_mod() - a2na3iimag)) * irot3.val();
}
if (s + 1 != (1 << (len - 2))) irot *= irate3[__builtin_ctzll(~(unsigned int)(s))];
}
len -= 2;
}
}
mint e = mint(n).inv();
for (auto& x : a) x *= e;
}
vector<mint> multiply(const vector<mint>& a, const vector<mint>& b) {
if (a.empty() || b.empty()) return vector<mint>();
int n = a.size(), m = b.size();
int sz = n + m - 1;
if (n <= 30 || m <= 30) {
if (n > 30) return multiply(b, a);
vector<mint> res(sz);
for (int i = 0; i < n; i++)
for (int j = 0; j < m; j++) res[i + j] += a[i] * b[j];
return res;
}
int sz1 = 1;
while (sz1 < sz) sz1 <<= 1;
vector<mint> res(sz1);
for (int i = 0; i < n; i++) res[i] = a[i];
ntt(res);
if (a == b)
for (int i = 0; i < sz1; i++) res[i] *= res[i];
else {
vector<mint> c(sz1);
for (int i = 0; i < m; i++) c[i] = b[i];
ntt(c);
for (int i = 0; i < sz1; i++) res[i] *= c[i];
}
intt(res);
res.resize(sz);
return res;
}
// c[i]=sum[j]a[j]b[i+j]
vector<mint> middle_product(const vector<mint>& a, const vector<mint>& b) {
if (b.empty() || a.size() > b.size()) return {};
int n = a.size(), m = b.size();
int sz = m - n + 1;
if (n <= 30 || sz <= 30) {
vector<mint> res(sz);
for (int i = 0; i < sz; i++)
for (int j = 0; j < n; j++) res[i] += a[j] * b[i + j];
return res;
}
int sz1 = 1;
while (sz1 < m) sz1 <<= 1;
vector<mint> res(sz1), b2(sz1);
reverse_copy(a.begin(), a.end(), res.begin());
copy(b.begin(), b.end(), b2.begin());
ntt(res);
ntt(b2);
for (int i = 0; i < res.size(); i++) res[i] *= b2[i];
intt(res);
res.resize(m);
res.erase(res.begin(), res.begin() + n - 1);
return res;
}
void ntt_doubling(vector<mint>& a) {
int n = (int)a.size();
auto b = a;
intt(b);
mint r = 1, zeta = mint(g).pow((mint::get_mod() - 1) / (n << 1));
for (int i = 0; i < n; i++) b[i] *= r, r *= zeta;
ntt(b);
copy(b.begin(), b.end(), back_inserter(a));
}
};
/**
* @brief NTT (数論変換)
* @docs docs/fft/ntt.md
*/
#line 2 "library/fps/formal-power-series.hpp"
template <class mint>
struct FormalPowerSeries : vector<mint> {
using vector<mint>::vector;
using FPS = FormalPowerSeries;
FormalPowerSeries(const vector<mint>& r) : vector<mint>(r) {}
FormalPowerSeries(vector<mint>&& r) : vector<mint>(std::move(r)) {}
FPS& operator=(const vector<mint>& r) {
vector<mint>::operator=(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 mint& 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 mint& r) {
if (this->empty()) this->resize(1);
(*this)[0] -= r;
return *this;
}
FPS& operator*=(const mint& 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;
if ((int)r.size() <= 64) {
FPS f(*this), g(r);
g.shrink();
mint coeff = g.at(g.size() - 1).inv();
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, mint(0));
return *this;
}
return *this = ((*this).rev().pre(n) * r.rev().inv(n)).pre(n).rev();
}
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 mint& v) const { return FPS(*this) += v; }
FPS operator-(const FPS& r) const { return FPS(*this) -= r; }
FPS operator-(const mint& v) const { return FPS(*this) -= v; }
FPS operator*(const FPS& r) const { return FPS(*this) *= r; }
FPS operator*(const mint& 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() == mint(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;
}
FPS pre(int sz) const {
return FPS(begin(*this), begin(*this) + min((int)this->size(), sz));
}
FPS operator>>=(int sz) {
assert(sz >= 0);
if ((int)this->size() <= sz)
this->clear();
else
this->erase(this->begin(), this->begin() + sz);
return *this;
}
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) {
assert(sz >= 0);
this->insert(this->begin(), sz, mint(0));
return *this;
}
FPS operator<<(int sz) const {
FPS ret(*this);
ret.insert(ret.begin(), sz, mint(0));
return ret;
}
FPS diff() const {
const int n = (int)this->size();
FPS ret(max(0, n - 1));
mint 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);
ret[0] = mint(0);
if (n > 0) ret[1] = mint(1);
auto mod = mint::get_mod();
for (int i = 2; i <= n; i++) ret[i] = (-ret[mod % i]) * (mod / i);
for (int i = 0; i < n; i++) ret[i + 1] *= (*this)[i];
return ret;
}
mint eval(mint x) const {
mint r = 0, w = 1;
for (auto& v : *this) r += w * v, w *= x;
return r;
}
FPS log(int deg = -1) const {
assert((*this)[0] == mint(1));
if (deg == -1) deg = (int)this->size();
return (this->diff() * this->inv(deg)).pre(deg - 1).integral();
}
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] != mint(0)) {
mint rev = mint(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, mint(0));
return ret;
}
if (__int128_t(i + 1) * k >= deg) return FPS(deg, mint(0));
}
return FPS(deg, mint(0));
}
static void* ntt_ptr;
static void set_ntt();
FPS& operator*=(const FPS& r);
FPS middle_product(const FPS& r) const;
void ntt();
void intt();
void ntt_doubling();
static int ntt_root();
FPS inv(int deg = -1) const;
FPS exp(int deg = -1) const;
};
template <typename mint>
void* FormalPowerSeries<mint>::ntt_ptr = nullptr;
#line 5 "library/fps/fps-ntt-friendly.hpp"
template <class mint>
void FormalPowerSeries<mint>::set_ntt() {
if (!ntt_ptr) ntt_ptr = new NTT<mint>;
}
template <class mint>
FormalPowerSeries<mint>& FormalPowerSeries<mint>::operator*=(const FormalPowerSeries<mint>& r) {
if (this->empty() || r.empty()) {
this->clear();
return *this;
}
set_ntt();
auto ret = static_cast<NTT<mint>*>(ntt_ptr)->multiply(*this, r);
return *this = FormalPowerSeries<mint>(ret.begin(), ret.end());
}
template <class mint>
FormalPowerSeries<mint> FormalPowerSeries<mint>::middle_product(const FormalPowerSeries<mint>& r) const {
set_ntt();
auto ret = static_cast<NTT<mint>*>(ntt_ptr)->middle_product(*this, r);
return FormalPowerSeries<mint>(ret.begin(), ret.end());
}
template <class mint>
void FormalPowerSeries<mint>::ntt() {
set_ntt();
static_cast<NTT<mint>*>(ntt_ptr)->ntt(*this);
}
template <class mint>
void FormalPowerSeries<mint>::intt() {
set_ntt();
static_cast<NTT<mint>*>(ntt_ptr)->intt(*this);
}
template <class mint>
void FormalPowerSeries<mint>::ntt_doubling() {
set_ntt();
static_cast<NTT<mint>*>(ntt_ptr)->ntt_doubling(*this);
}
template <typename mint>
int FormalPowerSeries<mint>::ntt_root() {
set_ntt();
return static_cast<NTT<mint>*>(ntt_ptr)->g;
}
template <typename mint>
FormalPowerSeries<mint> FormalPowerSeries<mint>::inv(int deg) const {
assert((*this)[0] != mint(0));
if (deg == -1) deg = (*this).size();
FPS ret{mint(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);
}
template <typename mint>
FormalPowerSeries<mint> FormalPowerSeries<mint>::exp(int deg) const {
assert((*this)[0] == mint(0));
if (deg == -1) deg = (*this).size();
FPS ret{mint(1)};
for (int i = 1; i < deg; i <<= 1)
ret = (ret * ((*this).pre(i << 1) - ret.log(i << 1) + 1)).pre(i << 1);
return ret.pre(deg);
}
#line 6 "main.cpp"
using fps = FormalPowerSeries<mint>;
#line 3 "library/fps/fps-rational.hpp"
template <class mint>
struct FPSRational {
using F = FormalPowerSeries<mint>;
using R = FPSRational;
F num, den;
FPSRational() : num(F{}), den(F{1}) {}
FPSRational(F f) : num(f), den(F{1}) {}
FPSRational(F f, F g) : num(f), den(g) {}
R& operator+=(const R& r) {
num *= r.den;
num += den * r.num;
den *= r.den;
return *this;
}
R& operator-=(const R& r) {
num *= r.den;
num -= den * r.num;
den *= r.den;
return *this;
}
R& operator*=(const R& r) {
num *= r.num;
den *= r.den;
return *this;
}
R& operator/=(const R& r) {
num *= r.den;
den *= r.num;
return *this;
}
R operator+(const R& r) const { return R(*this) += r; }
R operator-(const R& r) const { return R(*this) -= r; }
R operator*(const R& r) const { return R(*this) *= r; }
R operator/(const R& r) const { return R(*this) /= r; }
R inv() const { return {den, num}; }
F approx(int deg) const { return (den * num.inv(deg)).pre(deg); }
};
#line 8 "main.cpp"
using rat = FPSRational<mint>;
#line 3 "library/fps/bostan-mori.hpp"
// [x^n]f(x)/g(x)
template <class mint>
mint BostanMori(FormalPowerSeries<mint> f, FormalPowerSeries<mint> g, long long n) {
g.shrink();
mint ret = 0;
if (f.size() >= g.size()) {
auto q = f / g;
if (n < q.size()) ret += q[n];
f -= q * g;
f.shrink();
}
if (f.empty()) return ret;
FormalPowerSeries<mint>::set_ntt();
if (!FormalPowerSeries<mint>::ntt_ptr) {
f.resize(g.size() - 1);
for (; n > 0; n >>= 1) {
auto g1 = g;
for (int i = 1; i < g1.size(); i += 2) g1[i] = -g1[i];
auto p = f * g1, q = g * g1;
if (n & 1) {
for (int i = 0; i < f.size(); i++) f[i] = p[(i << 1) | 1];
} else {
for (int i = 0; i < f.size(); i++) f[i] = p[i << 1];
}
for (int i = 0; i < g.size(); i++) g[i] = q[i << 1];
}
return ret + f[0] / g[0];
} else {
int m = 1, log = 0;
while (m < g.size()) m <<= 1, log++;
mint wi = mint(FormalPowerSeries<mint>::ntt_root()).inv().pow((mint::get_mod() - 1) >> (log + 1));
vector<int> rev(m);
for (int i = 0; i < rev.size(); i++) rev[i] = (rev[i / 2] / 2) | ((i & 1) << (log - 1));
vector<mint> pow(m, 1);
for (int i = 1; i < m; i++) pow[rev[i]] = pow[rev[i - 1]] * wi;
f.resize(m), g.resize(m);
f.ntt(), g.ntt();
mint inv2 = mint(2).inv();
for (; n >= m; n >>= 1) {
f.ntt_doubling(), g.ntt_doubling();
if (n & 1) {
for (int i = 0; i < m; i++) f[i] = (f[i << 1] * g[(i << 1) | 1] - f[(i << 1) | 1] * g[i << 1]) * inv2 * pow[i];
} else {
for (int i = 0; i < m; i++) f[i] = (f[i << 1] * g[(i << 1) | 1] + f[(i << 1) | 1] * g[i << 1]) * inv2;
}
for (int i = 0; i < m; i++) g[i] = g[i << 1] * g[(i << 1) | 1];
f.resize(m), g.resize(m);
}
f.intt(), g.intt();
return ret + (f * g.inv())[n];
}
}
/**
* @brief Bostan-Mori
* @docs docs/fps/bostan-mori.md
*/
#line 10 "main.cpp"
rat RatSum(vector<rat> rs) {
for (int i = rs.size() - 1; i >= 1; i--)
rs[i - (i & -i)] += rs[i];
return rs[0];
}
const mint w = mint(6).pow((mint::get_mod() - 1) / 10);
void solve() {
int n, k, t;
in(n, k, t);
vector<int> a(n);
in(a);
vector r(k, vector<int>(n));
in(r);
mint invk = mint(k).inv();
vector<int> pow10(n + 1, 1);
rep(i, 0, n) pow10[i + 1] = pow10[i] * 10;
int m = pow10[n];
mint invm = mint(m).inv();
vector<mint> poww(100, 1);
rep(i, 1, poww.size()) poww[i] = poww[i - 1] * w;
vector<mint> z(m), za(m);
auto pre_calc = [&](auto rec, int k, int l, int r, vector<mint> f, mint va) -> void {
if (k < 0) {
z[l] = f[0];
za[l] = va;
return;
} else {
rep(t, 0, 10) {
int l1 = l + (r - l) / 10 * t;
int r1 = l1 + (r - l) / 10;
vector<mint> g(f.size() / 10);
rep(i, 0, g.size()) rep(j, 0, 10) {
g[i] += f[i + pow10[k] * j] * poww[j * t];
}
rec(rec, k - 1, l1, r1, g, va * poww[a[k] * t]);
}
}
};
{
vector<mint> f0(m);
rep(i, 0, k) {
int x = 0;
rep(j, 0, n) x += pow10[j] * r[i][j];
f0[x] = invk;
}
pre_calc(pre_calc, n - 1, 0, m, f0, 1);
}
rat f;
{
vector<rat> rs;
rep(i, 0, m) rs.push_back(rat(fps{invm}, fps{za[i], -za[i] * z[i]}));
f = RatSum(rs);
}
{
vector<rat> rs;
rep(i, 0, m) rs.push_back(rat(fps{invm}, fps{1, -z[i]}));
f /= RatSum(rs);
}
f /= rat(fps{1, -1});
mint ans = BostanMori<mint>(f.num, f.den, t);
out(ans);
}
int main() {
int t = 1;
// in(t);
while (t--) solve();
}