結果

問題 No.2963 Mecha DESU
ユーザー 👑 bo9chanbo9chan
提出日時 2024-11-16 17:42:59
言語 C++23
(gcc 12.3.0 + boost 1.83.0)
結果
AC  
実行時間 511 ms / 2,000 ms
コード長 48,914 bytes
コンパイル時間 3,052 ms
コンパイル使用メモリ 186,548 KB
実行使用メモリ 36,476 KB
最終ジャッジ日時 2024-11-16 17:43:19
合計ジャッジ時間 18,290 ms
ジャッジサーバーID
（参考情報） 		judge1 / judge3
このコードへのチャレンジ
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テストケース

テストケース表示
入力 結果 実行時間
実行使用メモリ
testcase_00 AC 2 ms
5,248 KB
testcase_01 AC 1 ms
5,248 KB
testcase_02 AC 2 ms
5,248 KB
testcase_03 AC 2 ms
5,248 KB
testcase_04 AC 2 ms
5,248 KB
testcase_05 AC 2 ms
5,248 KB
testcase_06 AC 2 ms
5,248 KB
testcase_07 AC 2 ms
5,248 KB
testcase_08 AC 2 ms
5,248 KB
testcase_09 AC 2 ms
5,248 KB
testcase_10 AC 2 ms
5,248 KB
testcase_11 AC 2 ms
5,248 KB
testcase_12 AC 2 ms
5,248 KB
testcase_13 AC 485 ms
36,348 KB
testcase_14 AC 456 ms
36,348 KB
testcase_15 AC 479 ms
36,472 KB
testcase_16 AC 479 ms
36,476 KB
testcase_17 AC 450 ms
36,476 KB
testcase_18 AC 475 ms
36,476 KB
testcase_19 AC 511 ms
36,476 KB
testcase_20 AC 274 ms
23,408 KB
testcase_21 AC 66 ms
7,424 KB
testcase_22 AC 44 ms
7,040 KB
testcase_23 AC 30 ms
5,248 KB
testcase_24 AC 295 ms
24,148 KB
testcase_25 AC 307 ms
25,516 KB
testcase_26 AC 406 ms
33,444 KB
testcase_27 AC 191 ms
17,776 KB
testcase_28 AC 387 ms
29,992 KB
testcase_29 AC 273 ms
22,904 KB
testcase_30 AC 22 ms
5,248 KB
testcase_31 AC 129 ms
12,560 KB
testcase_32 AC 58 ms
7,936 KB
testcase_33 AC 323 ms
27,900 KB
testcase_34 AC 326 ms
26,564 KB
testcase_35 AC 115 ms
10,752 KB
testcase_36 AC 340 ms
29,220 KB
testcase_37 AC 46 ms
6,016 KB
testcase_38 AC 115 ms
12,288 KB
testcase_39 AC 342 ms
28,732 KB
testcase_40 AC 199 ms
18,688 KB
testcase_41 AC 351 ms
28,556 KB
testcase_42 AC 269 ms
22,848 KB
testcase_43 AC 461 ms
36,076 KB
testcase_44 AC 424 ms
35,868 KB
testcase_45 AC 367 ms
31,068 KB
testcase_46 AC 183 ms
17,012 KB
testcase_47 AC 301 ms
24,364 KB
testcase_48 AC 99 ms
10,496 KB
testcase_49 AC 263 ms
23,132 KB
testcase_50 AC 482 ms
36,472 KB
testcase_51 AC 449 ms
36,412 KB
testcase_52 AC 476 ms
36,472 KB
testcase_53 AC 474 ms
36,348 KB
testcase_54 AC 430 ms
36,476 KB
testcase_55 AC 2 ms
5,248 KB
testcase_56 AC 2 ms
5,248 KB
testcase_57 AC 379 ms
35,696 KB
testcase_58 AC 30 ms
5,248 KB
testcase_59 AC 2 ms
5,248 KB
権限があれば一括ダウンロードができます

ソースコード

diff # 

#ifdef __LOCAL
#include <mytemplate.hpp> // プリコンパイル済みヘッダ ~/local/include/mytemplate.hpp.gch
#else
#include <algorithm>
#include <array>
#include <bitset>
#include <cassert>
#include <climits>
#include <cmath>
#include <deque>
#include <functional>
#include <iomanip>
#include <iostream>
#include <iterator>
#include <list>
#include <map>
#include <memory>
#include <numeric>
#include <optional>
#include <queue>
#include <random>
#include <set>
#include <stack>
#include <string>
#include <unordered_map>
#include <unordered_set>
#include <utility>
#include <vector>
#endif
using namespace std;
// #pragma GCC target "no-avx"  // gcc12.2のバグ回避用。boost使用時はコメントアウトの事
// #include <boost/multiprecision/cpp_int.hpp>
// using bint = boost::multiprecision::cpp_int;
using ll = long long;
using vl = vector<ll>;
using vvl = vector<vl>;
using vb = vector<bool>;
using vvb = vector<vb>;
using vd = vector<double>;
using vvd = vector<vd>;
using vc = vector<char>;
using vvc = vector<vc>;
using pll = pair<ll, ll>;
template <class T> using min_priority_queue = priority_queue<T, vector<T>, greater<T>>;
#define REP1(i, n) REP3(i, 0, n, 1)
#define REP2(i, s, n) REP3(i, s, n, 1)
#define REP3(i, s, n, d) for (ll i = (ll)(s); i < (ll)(n); i += (d))
#define REP_OVERLOAD(e1, e2, e3, e4, NAME, ...) NAME
#define rep(...) REP_OVERLOAD(__VA_ARGS__, REP3, REP2, REP1)(__VA_ARGS__)
#define DEP1(i, n) DEP3(i, n, -1, 1)
#define DEP2(i, n, s) DEP3(i, n, s, 1)
#define DEP3(i, n, s, d) for (ll i = (ll)(n); (ll)(s) < i; i -= (d))
#define DEP_OVERLOAD(e1, e2, e3, e4, NAME, ...) NAME
#define dep(...) DEP_OVERLOAD(__VA_ARGS__, DEP3, DEP2, DEP1)(__VA_ARGS__)
#define fore(e, a) for (auto&& e : (a))
#define len(a) (ll)(a).size()
#define all(a) (a).begin(), (a).end()
#define rall(a) (a).rbegin(), (a).rend()
#define pb push_back
#define eb emplace_back
#define fi first
#define se second
#define th third
#define fo fourth
/*********    C O N S T A N T    **********/
constexpr ll INF = LONG_LONG_MAX / 2 - 10000LL; // 4,611,686,018,427,377,903  ~= 4.6e+18, 19桁
const double PI = acos(-1);
/*********    1 2 8 b i t 整 数  **********/
using int128 = __int128_t;
istream& operator>>(istream& stream, int128& val);
ostream& operator<<(ostream& stream, const int128& val_);
/*********       P A I R       **********/
// 単項演算子（+ -）
template <class T, class U> pair<T, U> operator+(pair<T, U> a);
template <class T, class U> pair<T, U> operator-(pair<T, U> a);
// 複合代入演算子
template <class T, class U> pair<T, U>& operator+=(pair<T, U>& a, const pair<T, U> b);
template <class T, class U> pair<T, U>& operator-=(pair<T, U>& a, const pair<T, U> b);
template <class T, class U> pair<T, U>& operator*=(pair<T, U>& a, const pair<T, U> b);
template <class T, class U> pair<T, U>& operator/=(pair<T, U>& a, const pair<T, U> b);
template <class T, class U> pair<T, U>& operator%=(pair<T, U>& a, const pair<T, U> b);
template <class T, class U, class V> pair<T, U>& operator+=(pair<T, U>& a, const V b);
template <class T, class U, class V> pair<T, U>& operator-=(pair<T, U>& a, const V b);
template <class T, class U, class V> pair<T, U>& operator*=(pair<T, U>& a, const V b);
template <class T, class U, class V> pair<T, U>& operator/=(pair<T, U>& a, const V b);
template <class T, class U, class V> pair<T, U>& operator%=(pair<T, U>& a, const V b);
// 算術演算子
template <class T, class U, class V> pair<T, U> operator+(pair<T, U> a, const V& b);
template <class T, class U, class V> pair<T, U> operator-(pair<T, U> a, const V& b);
template <class T, class U, class V> pair<T, U> operator*(pair<T, U> a, const V& b);
template <class T, class U, class V> pair<T, U> operator/(pair<T, U> a, const V& b);
template <class T, class U, class V> pair<T, U> operator%(pair<T, U> a, const V& b);
// 入出力
template <class T, class U> istream& operator>>(istream& stream, pair<T, U>& a);
template <class T, class U> ostream& operator<<(ostream& stream, const pair<T, U>& a);
/*********       V E C T O R      **********/
// 単項演算子（+ -）
template <class T> vector<T> operator+(vector<T> a);
template <class T> vector<T> operator-(vector<T> a);
// 複合代入演算子
template <class T> vector<T>& operator+=(vector<T>& a, const vector<T> b);
template <class T> vector<T>& operator-=(vector<T>& a, const vector<T> b);
template <class T> vector<T>& operator*=(vector<T>& a, const vector<T> b);
template <class T> vector<T>& operator/=(vector<T>& a, const vector<T> b);
template <class T> vector<T>& operator%=(vector<T>& a, const vector<T> b);
template <class T, class U> vector<T>& operator+=(vector<T>& a, const U b);
template <class T, class U> vector<T>& operator-=(vector<T>& a, const U b);
template <class T, class U> vector<T>& operator*=(vector<T>& a, const U b);
template <class T, class U> vector<T>& operator/=(vector<T>& a, const U b);
template <class T, class U> vector<T>& operator%=(vector<T>& a, const U b);
// 算術演算子
template <class T, class U> vector<T> operator+(vector<T> a, const U& b);
template <class T, class U> vector<T> operator-(vector<T> a, const U& b);
template <class T, class U> vector<T> operator*(vector<T> a, const U& b);
template <class T, class U> vector<T> operator/(vector<T> a, const U& b);
template <class T, class U> vector<T> operator%(vector<T> a, const U& b);
// 入出力
template <class T> istream& operator>>(istream& stream, vector<T>& a);
template <class T> ostream& operator<<(ostream& stream, const vector<T>& v);
template <class T> ostream& operator<<(ostream& stream, const vector<vector<T>>& vv);
// 計算
template <class T> T Sum(const vector<T>& v);
template <class T> T Max(const vector<T>& v);
template <class T> T Min(const vector<T>& v);
template <class T> ll Argmax(const vector<T>& v);
template <class T> ll Argmin(const vector<T>& v);
template <class T, class U> bool Contains(const vector<T>& v, const U& a);
template <class T> void Unique(vector<T>& v);
// 生成
template <class T, class U> vector<T> make_vector(int n, U v);
template <class T, class... Args> auto make_vector(int n, Args... args) {
    auto val = make_vector<T>(args...);
    return make_vector<decltype(val)>(n, move(val));
}
/*********        S E T        **********/
template <class T> istream& operator>>(istream& stream, set<T>& st);
template <class T> ostream& operator<<(ostream& stream, const set<T>& st);
template <class T> T Max(const set<T>& st);
template <class T> T Min(const set<T>& st);
/*********   M U L T I S E T   **********/
template <class T> istream& operator>>(istream& stream, multiset<T>& st);
template <class T> ostream& operator<<(ostream& stream, const multiset<T>& st);
template <class T> T Max(const multiset<T>& st);
template <class T> T Min(const multiset<T>& st);
template <class T> bool EraseOne(multiset<T>& st, const T& x);
/*********       C H A R       **********/
bool IsUpper(const char& c);
bool IsLower(const char& c);
char ToUpper(const char& c);
char ToLower(const char& c);
/*********     S T R I N G     **********/
string ToUpper(const string& s);
string ToLower(const string& s);
bool IsPalindrome(const string& S);
/*********       T R I O       **********/
template <class T1, class T2, class T3> struct trio {
    T1 first;
    T2 second;
    T3 third;
    // コンストラクタ・代入演算子
    trio();
    trio(const T1& x);
    trio(const T1& x, const T2& y, const T3& z);
    trio(const trio& t);
    trio& operator=(const trio& t);
    // 比較演算子
    bool operator<(const trio& t) const;
    bool operator==(const trio& t) const;
    bool operator!=(const trio& t) const;
    bool operator>(const trio& t) const;
    bool operator<=(const trio& t) const;
    bool operator>=(const trio& t) const;
    // 単項演算子（+ -）
    trio operator+() const;
    trio operator-() const;
    // 入出力
    template <class U1, class U2, class U3> friend istream& operator>>(istream&, trio<U1, U2, U3>&);
    template <class U1, class U2, class U3> friend ostream& operator<<(ostream&, const trio<U1, U2, U3>&);
    // `tie(a, b, c) = tll{3, 5, 7}`をするのに必要（コンパイラが警告？を出すのでここで定義）
    operator tuple<T1&, T2&, T3&>() {
        return tie(first, second, third);
    }
};
// 複合代入演算子
template <class T1, class T2, class T3>
trio<T1, T2, T3>& operator+=(trio<T1, T2, T3>& a, const trio<T1, T2, T3> b);
template <class T1, class T2, class T3>
trio<T1, T2, T3>& operator-=(trio<T1, T2, T3>& a, const trio<T1, T2, T3> b);
template <class T1, class T2, class T3>
trio<T1, T2, T3>& operator*=(trio<T1, T2, T3>& a, const trio<T1, T2, T3> b);
template <class T1, class T2, class T3>
trio<T1, T2, T3>& operator/=(trio<T1, T2, T3>& a, const trio<T1, T2, T3> b);
template <class T1, class T2, class T3>
trio<T1, T2, T3>& operator%=(trio<T1, T2, T3>& a, const trio<T1, T2, T3> b);
template <class T1, class T2, class T3, class U> trio<T1, T2, T3>& operator+=(trio<T1, T2, T3>& a, const U b);
template <class T1, class T2, class T3, class U> trio<T1, T2, T3>& operator-=(trio<T1, T2, T3>& a, const U b);
template <class T1, class T2, class T3, class U> trio<T1, T2, T3>& operator*=(trio<T1, T2, T3>& a, const U b);
template <class T1, class T2, class T3, class U> trio<T1, T2, T3>& operator/=(trio<T1, T2, T3>& a, const U b);
template <class T1, class T2, class T3, class U> trio<T1, T2, T3>& operator%=(trio<T1, T2, T3>& a, const U b);
// 算術演算子
template <class T1, class T2, class T3, class U> trio<T1, T2, T3> operator+(trio<T1, T2, T3> a, const U& b);
template <class T1, class T2, class T3, class U> trio<T1, T2, T3> operator-(trio<T1, T2, T3> a, const U& b);
template <class T1, class T2, class T3, class U> trio<T1, T2, T3> operator*(trio<T1, T2, T3> a, const U& b);
template <class T1, class T2, class T3, class U> trio<T1, T2, T3> operator/(trio<T1, T2, T3> a, const U& b);
template <class T1, class T2, class T3, class U> trio<T1, T2, T3> operator%(trio<T1, T2, T3> a, const U& b);
using tll = trio<ll, ll, ll>;
/*********    Q U A R T E T    **********/
template <class T1, class T2, class T3, class T4> struct quartet {
    T1 first;
    T2 second;
    T3 third;
    T4 fourth;
    // コンストラクタ・代入演算子
    quartet();
    quartet(const T1& x);
    quartet(const T1& x, const T2& y, const T3& z, const T4& w);
    quartet(const quartet& t);
    quartet& operator=(const quartet& t);
    // 比較演算子
    bool operator<(const quartet& t) const;
    bool operator==(const quartet& t) const;
    bool operator!=(const quartet& other) const;
    bool operator>(const quartet& other) const;
    bool operator<=(const quartet& other) const;
    bool operator>=(const quartet& other) const;
    // 単項演算子（+ -）
    quartet operator+() const;
    quartet operator-() const;
    // 入出力
    template <class U1, class U2, class U3, class U4>
    friend istream& operator>>(istream&, quartet<U1, U2, U3, U4>&);
    template <class U1, class U2, class U3, class U4>
    friend ostream& operator<<(ostream&, const quartet<U1, U2, U3, U4>&);
    // `tie(a, b, c, d) = qll{3, 5, 7, 9}`をするのに必要（コンパイラが警告？を出すのでここで定義）
    operator tuple<T1&, T2&, T3&, T4&>() {
        return tie(first, second, third, fourth);
    }
};
// 複合代入演算子
template <class T1, class T2, class T3, class T4>
quartet<T1, T2, T3, T4>& operator+=(quartet<T1, T2, T3, T4>& a, const quartet<T1, T2, T3, T4> b);
template <class T1, class T2, class T3, class T4>
quartet<T1, T2, T3, T4>& operator-=(quartet<T1, T2, T3, T4>& a, const quartet<T1, T2, T3, T4> b);
template <class T1, class T2, class T3, class T4>
quartet<T1, T2, T3, T4>& operator*=(quartet<T1, T2, T3, T4>& a, const quartet<T1, T2, T3, T4> b);
template <class T1, class T2, class T3, class T4>
quartet<T1, T2, T3, T4>& operator/=(quartet<T1, T2, T3, T4>& a, const quartet<T1, T2, T3, T4> b);
template <class T1, class T2, class T3, class T4>
quartet<T1, T2, T3, T4>& operator%=(quartet<T1, T2, T3, T4>& a, const quartet<T1, T2, T3, T4> b);
template <class T1, class T2, class T3, class T4, class U>
quartet<T1, T2, T3, T4>& operator+=(quartet<T1, T2, T3, T4>& a, const U b);
template <class T1, class T2, class T3, class T4, class U>
quartet<T1, T2, T3, T4>& operator-=(quartet<T1, T2, T3, T4>& a, const U b);
template <class T1, class T2, class T3, class T4, class U>
quartet<T1, T2, T3, T4>& operator*=(quartet<T1, T2, T3, T4>& a, const U b);
template <class T1, class T2, class T3, class T4, class U>
quartet<T1, T2, T3, T4>& operator/=(quartet<T1, T2, T3, T4>& a, const U b);
template <class T1, class T2, class T3, class T4, class U>
quartet<T1, T2, T3, T4>& operator%=(quartet<T1, T2, T3, T4>& a, const U b);
// 算術演算子
template <class T1, class T2, class T3, class T4, class U>
quartet<T1, T2, T3, T4> operator+(quartet<T1, T2, T3, T4> a, const U& b);
template <class T1, class T2, class T3, class T4, class U>
quartet<T1, T2, T3, T4> operator-(quartet<T1, T2, T3, T4> a, const U& b);
template <class T1, class T2, class T3, class T4, class U>
quartet<T1, T2, T3, T4> operator*(quartet<T1, T2, T3, T4> a, const U& b);
template <class T1, class T2, class T3, class T4, class U>
quartet<T1, T2, T3, T4> operator/(quartet<T1, T2, T3, T4> a, const U& b);
template <class T1, class T2, class T3, class T4, class U>
quartet<T1, T2, T3, T4> operator%(quartet<T1, T2, T3, T4> a, const U& b);
using qll = quartet<ll, ll, ll, ll>;
/*********     R A N D O M     **********/
struct Random {
    mt19937_64 rnd;
    Random();
    Random(ll seed);
    ll randint(ll a, ll b);
    double randreal(double a, double b);
    char randchar(char a, char b);
    template <class T> void shuffle(vector<T>& vec);
};
/*********       T I M E R       **********/
struct Timer {
    int time_limit;
    int start_time;
    Timer(double time_limit_sec);
    bool is_time_up() const;
    double get_elapsed_ratio() const;
};
/*********       P R I N T       **********/
template <class T> void print(const T& e);
template <class H, class... T> void print(const H& h, const T&... t);
template <class... T> void End(const T&... t);
/*********       D E B U G       **********/
#ifdef __LOCAL
#define debug(...)                                                                                           \
    if (DEBUG) do {                                                                                          \
            cout << '[' << #__VA_ARGS__ << "] ";                                                             \
            debug_(__VA_ARGS__);                                                                             \
    } while (0)
#else
#define debug(...)
#endif
void dbg_(const long long& e);
template <class T> void dbg_(const T& e);
template <class T, class U> void dbg_(const pair<T, U>& p);
template <class T1, class T2, class T3> void dbg_(const trio<T1, T2, T3>& t);
template <class T1, class T2, class T3, class T4> void dbg_(const quartet<T1, T2, T3, T4>& t);
template <class T> void debug_(const T& e);
template <class T> void debug_(const vector<T>& v);
template <class T> void debug_(const vector<vector<T>>& vv);
template <class T, class U> void debug_(const map<T, U>& mp);
template <class T, class U> void debug_(const vector<map<T, U>>& vm);
template <class T> void debug_(const set<T>& st);
template <class T> void debug_(const multiset<T>& st);
template <class T> void debug_(const vector<set<T>>& vs);
template <class T> void debug_(const vector<multiset<T>>& vs);
template <class H, class... T> void debug_(const H& h, const T&... t);
/*********       O T H E R S       **********/
template <class T, class U> bool chmin(T& a, U b);
template <class T, class U> bool chmax(T& a, U b);
template <class T, class U> T Mod(const T& a, const U& m);
template <class T, class U> T Ceil(const T& x, const U& y);
template <class T, class U> T Floor(const T& x, const U& y);
template <class T> ll Sign(const T& x);
ll Isqrt(ll n);
ll Comb(ll n, ll r);
template <class T> T Pow(T x, ll n);
template <class T> T Pow(T x, ll n, T mod);
template <class T> T Aseries(T a, T d, ll n);
template <class T> T Gseries(T a, T r, ll n);
bool Bit(ll b, int i);
ll Popcount(ll b);
ll Mask(ll n);
ll Mask(ll n, ll m);
pll GetOverlap(ll l1, ll r1, ll l2, ll r2);
bool IsOverlap(ll l1, ll r1, ll l2, ll r2);
void PrintYesNo(bool b);
/*********       M  O  D       **********/
#include <atcoder/modint>  // https://atcoder.github.io/ac-library/production/document_ja/
using namespace atcoder;
using mint = modint998244353;     // modint1000000007;
istream& operator >>(istream& stream, mint& e) {ll n; stream >> n; e = n; return stream; }
ostream& operator <<(ostream& stream, const mint& e) { stream << e.val(); return stream; }
using vm = vector<mint>;
using vvm = vector<vm>;
/***************************************/
constexpr bool DEBUG = true;


// 線形篩　参考：https://atcoder.jp/contests/abc292/editorial/5912
struct LinearSieve {
    ll N;
    vl primes;           // N以下の素数列
    vl spf;              // smallest prime factor (ex. spf[75] = 3)
    vl power;            // iが含むspf[i]の個数 (ex. power[40] = 3)
    vl prev_divisor;     // prev_divisor[i] = i / spf[i]^power[i].

    LinearSieve(ll N) : N(N), spf(N + 1), power(N + 1), prev_divisor(N + 1) {
        rep(i, 2, N + 1) {
            if (spf[i] == 0) {
                primes.pb(i);
                spf[i] = i;
                power[i] = 1;
                prev_divisor[i] = 1;
            }
            fore(p, primes) {
                if (p * i > N) break;
                spf[p * i] = p;
                if (p == spf[i]) {
                    prev_divisor[p * i] = prev_divisor[i];
                    power[p * i] = power[i] + 1;
                    break;
                }
                prev_divisor[p * i] = i;
                power[p * i] = 1;
            }
        }
    }

    // 素因数分解結果を取得, O(mlogm) (m: nの素因数の種類数 )
    map<ll, ll> get(ll n) {
        assert(1 <= n && n <= N);
        map<ll, ll> mp;
        while(n > 1) {
            mp[spf[n]] = power[n];
            n = prev_divisor[n];
        }
        return mp;
    }

    // 約数列挙（ソートなし）, O(M) (M: nの約数の個数)
    //参考: https://atcoder.jp/contests/abc170/submissions/14359259
    vl divisor(ll n) {
        // 約数の個数を求め、配列を初期化
        ll cnt = divisor_count(n);
        vl ret(cnt);
        ret[0] = 1;
        cnt = 1;  // 計算した約数の個数

        while(n > 1) {
            ll d = spf[n];
            rep(k, 1, power[n] + 1) {
                rep(j, cnt) {
                    ret[k * cnt + j] = ret[j] * d;
                }
                d *= spf[n];
            }
            cnt *= power[n] + 1;
            n = prev_divisor[n];
        }
        return ret;
    }

    // 約数の個数
    ll divisor_count(ll n) {
        assert(1 <= n && n <= N);
        ll ret = 1;
        while(n > 1) {
            ret *= power[n] + 1;
            n = prev_divisor[n];
        }
        return ret;
    }

    // オイラーのφ関数
    ll phi(ll n) {
        assert(1 <= n && n <= N);
        ll ret = n;
        while(n > 1) {
            ll p = spf[n];
            ret /= p;
            ret *= p - 1;
            n = prev_divisor[n];
        }
        return ret;
    }
};
// LinearSieve spf(100);          // O(N)
// auto mp = spf.get(12);         // (2, 2), (3, 1). 素因数分解, O(mlogm) (m: 素因数の種類数)
// vl vec = spf.divisor(12);      // {1, 2, 4, 3, 6, 12}. 約数列挙（ソートなし）, O(M) (M: nの約数の個数)
// ll n = spf.divisor_count(12);  // 6. 約数の個数, O(m)
// ll p = spf.phi(12);            // p = 4. オイラーのφ関数. 12と互いに素: {1, 5, 7, 11}



int main() {
    cin.tie(nullptr);
    cout << fixed << setprecision(10);

    ll N, M, K;
    cin >> N >> M >> K;
    vl A(M);
    cin >> A;

    vl C(N + 1);
    fore(a, A) {
        if (a > N) continue;
        C[a]++;
    }

    mint ans = 0;
    auto spf = LinearSieve(N);
    rep(i, 1, N + 1) {
        vl vec = spf.divisor(i);
        ll rest = M;
        fore(e, vec) {
            rest -= C[e];
        }
        ans += 1 - Pow((mint)rest / M, K);
    }
    print(ans);
}



/*********    1 2 8 b i t 整 数    *********/
// 128bit整数を入力. LONG_LONG_MAX超過/LONG_LONG_MIN未満もOK.
istream& operator>>(istream& stream, int128& val) {
    string s;
    stream >> s;
    assert(s.size() > 0);

    val = 0;
    bool is_minus = false;
    for (int i = 0; i < (int)s.size(); i++) {
        if (i == 0 && s[i] == '-') {
            assert(s.size() > 1);
            is_minus = true;
        } else {
            assert('0' <= s[i] && s[i] <= '9');
            val *= 10;
            val += s[i] - '0';
        }
    }
    if (is_minus) val *= -1;
    return stream;
}
// 128bit整数を出力.
ostream& operator<<(ostream& stream, const int128& val_) {
    auto val(val_);
    if (val == 0) return stream << 0;
    if (val < 0) {
        stream << '-';
        val *= -1;
    }
    string s;
    while (val > 0) {
        s += (val % 10) + '0';
        val /= 10;
    }
    reverse(all(s));
    return stream << s;
}
/*********       P A I R       **********/
/* 単項演算子（+ -） */
template <class T, class U> pair<T, U> operator+(pair<T, U> a) {
    return a;
}
template <class T, class U> pair<T, U> operator-(pair<T, U> a) {
    return a * (-1);
}
/* 複合代入演算子 (pair) */
template <class T, class U> pair<T, U>& operator+=(pair<T, U>& a, const pair<T, U> b) {
    a.fi += b.fi;
    a.se += b.se;
    return a;
}
template <class T, class U> pair<T, U>& operator-=(pair<T, U>& a, const pair<T, U> b) {
    a.fi -= b.fi;
    a.se -= b.se;
    return a;
}
template <class T, class U> pair<T, U>& operator*=(pair<T, U>& a, const pair<T, U> b) {
    a.fi *= b.fi;
    a.se *= b.se;
    return a;
}
template <class T, class U> pair<T, U>& operator/=(pair<T, U>& a, const pair<T, U> b) {
    a.fi /= b.fi;
    a.se /= b.se;
    return a;
}
template <class T, class U> pair<T, U>& operator%=(pair<T, U>& a, const pair<T, U> b) {
    a.fi %= b.fi;
    a.se %= b.se;
    return a;
}
template <class T, class U, class V> pair<T, U>& operator+=(pair<T, U>& a, const V b) {
    a.fi += b;
    a.se += b;
    return a;
}
template <class T, class U, class V> pair<T, U>& operator-=(pair<T, U>& a, const V b) {
    a.fi -= b;
    a.se -= b;
    return a;
}
template <class T, class U, class V> pair<T, U>& operator*=(pair<T, U>& a, const V b) {
    a.fi *= b;
    a.se *= b;
    return a;
}
template <class T, class U, class V> pair<T, U>& operator/=(pair<T, U>& a, const V b) {
    a.fi /= b;
    a.se /= b;
    return a;
}
template <class T, class U, class V> pair<T, U>& operator%=(pair<T, U>& a, const V b) {
    a.fi %= b;
    a.se %= b;
    return a;
}
/* 算術演算子 (pair) */
template <class T, class U, class V> pair<T, U> operator+(pair<T, U> a, const V& b) {
    return a += b;
}
template <class T, class U, class V> pair<T, U> operator-(pair<T, U> a, const V& b) {
    return a -= b;
}
template <class T, class U, class V> pair<T, U> operator*(pair<T, U> a, const V& b) {
    return a *= b;
}
template <class T, class U, class V> pair<T, U> operator/(pair<T, U> a, const V& b) {
    return a /= b;
}
template <class T, class U, class V> pair<T, U> operator%(pair<T, U> a, const V& b) {
    return a %= b;
}
// 入出力
template <class T, class U> istream& operator>>(istream& stream, pair<T, U>& a) {
    stream >> a.fi >> a.se;
    return stream;
}
template <class T, class U> ostream& operator<<(ostream& stream, const pair<T, U>& a) {
    stream << a.fi << " " << a.se;
    return stream;
}

/*********       V E C T O R      **********/
/* 単項演算子（+ -） */
template <class T> vector<T> operator+(vector<T> a) {
    return a;
}
template <class T> vector<T> operator-(vector<T> a) {
    return a * (-1);
}
/* 複合代入演算子 (vector) */
template <class T> vector<T>& operator+=(vector<T>& a, const vector<T> b) {
    assert(a.size() == b.size());
    rep(i, 0, a.size()) a[i] += b[i];
    return a;
}
template <class T> vector<T>& operator-=(vector<T>& a, const vector<T> b) {
    assert(a.size() == b.size());
    rep(i, 0, a.size()) a[i] -= b[i];
    return a;
}
template <class T> vector<T>& operator*=(vector<T>& a, const vector<T> b) {
    assert(a.size() == b.size());
    rep(i, 0, a.size()) a[i] *= b[i];
    return a;
}
template <class T> vector<T>& operator/=(vector<T>& a, const vector<T> b) {
    assert(a.size() == b.size());
    rep(i, 0, a.size()) a[i] /= b[i];
    return a;
}
template <class T> vector<T>& operator%=(vector<T>& a, const vector<T> b) {
    assert(a.size() == b.size());
    rep(i, 0, a.size()) a[i] %= b[i];
    return a;
}
template <class T, class U> vector<T>& operator+=(vector<T>& a, const U b) {
    fore(e, a) e += b;
    return a;
}
template <class T, class U> vector<T>& operator-=(vector<T>& a, const U b) {
    fore(e, a) e -= b;
    return a;
}
template <class T, class U> vector<T>& operator*=(vector<T>& a, const U b) {
    fore(e, a) e *= b;
    return a;
}
template <class T, class U> vector<T>& operator/=(vector<T>& a, const U b) {
    fore(e, a) e /= b;
    return a;
}
template <class T, class U> vector<T>& operator%=(vector<T>& a, const U b) {
    fore(e, a) e %= b;
    return a;
}
/* 算術演算子 (vector) */
template <class T, class U> vector<T> operator+(vector<T> a, const U& b) {
    return a += b;
}
template <class T, class U> vector<T> operator-(vector<T> a, const U& b) {
    return a -= b;
}
template <class T, class U> vector<T> operator*(vector<T> a, const U& b) {
    return a *= b;
}
template <class T, class U> vector<T> operator/(vector<T> a, const U& b) {
    return a /= b;
}
template <class T, class U> vector<T> operator%(vector<T> a, const U& b) {
    return a %= b;
}
/* 入出力 (vector) */
// ベクトルを入力
template <class T> istream& operator>>(istream& stream, vector<T>& a) {
    fore(e, a) stream >> e;
    return stream;
}
// ベクトルを空白区切りで出力
template <class T> ostream& operator<<(ostream& stream, const vector<T>& v) {
    if (v.size()) {
        stream << v[0];
        rep(i, 1, v.size()) cout << " " << v[i];
    }
    return stream;
}
// 2次元ベクトルを空白/改行区切りで出力
template <class T> ostream& operator<<(ostream& stream, const vector<vector<T>>& vv) {
    if (vv.size()) {
        stream << vv[0];
        rep(i, 1, vv.size()) cout << '\n' << vv[i];
    }
    return stream;
}
/* 計算 (vector) */
// 合計。v.size() == 0 のとき T() を返す
template <class T> T Sum(const vector<T>& v) {
    return reduce(all(v));
}
template <class T> T Max(const vector<T>& v) {
    assert(v.size());
    return *max_element(all(v));
}
template <class T> T Min(const vector<T>& v) {
    assert(v.size());
    return *min_element(all(v));
}
// 最大要素を指す最初の位置を返す（0-origin）, O(N).
template <class T> ll Argmax(const vector<T>& v) {
    assert(v.size());
    return max_element(all(v)) - v.begin();
}
// 最小要素を指す最初の位置を返す（0-origin）, O(N).
template <class T> ll Argmin(const vector<T>& v) {
    assert(v.size());
    return min_element(all(v)) - v.begin();
}
// 要素`a`がベクトル`v`に1つ以上含まれるかどうか
template <class T, class U> bool Contains(const vector<T>& v, const U& a) {
    return find(all(v), a) != v.end();
}
// ソートされたユニーク値のベクトルにinplaceで書き換える, O(NlogN).
template <class T> void Unique(vector<T>& v) {
    sort(all(v));
    v.erase(unique(all(v)), v.end());
}
/* 生成 (vector) */
template <class T, class U> vector<T> make_vector(int n, U v) {
    return vector<T>(n, v);
}

/*********        S E T        **********/
// 一要素を入力
template <class T> istream& operator>>(istream& stream, set<T>& st) {
    T e;
    stream >> e;
    st.insert(e);
    return stream;
}
// 空白区切りで出力
template <class T> ostream& operator<<(ostream& stream, const set<T>& st) {
    if (st.size()) {
        auto it = st.begin();
        stream << *it++;
        for (; it != st.end(); it++)
            cout << " " << *it;
    }
    return stream;
}
template <class T> T Max(const set<T>& st) {
    assert(st.size());
    return *prev(st.end());
}
template <class T> T Min(const set<T>& st) {
    assert(st.size());
    return *st.begin();
}

/*********   M U L T I S E T   **********/
// 一要素を入力
template <class T> istream& operator>>(istream& stream, multiset<T>& st) {
    T e;
    stream >> e;
    st.insert(e);
    return stream;
}
// 空白区切りで出力
template <class T> ostream& operator<<(ostream& stream, const multiset<T>& st) {
    if (st.size()) {
        auto it = st.begin();
        stream << *it++;
        for (; it != st.end(); it++)
            cout << " " << *it;
    }
    return stream;
}
template <class T> T Max(const multiset<T>& st) {
    assert(st.size());
    return *prev(st.end());
}
template <class T> T Min(const multiset<T>& st) {
    assert(st.size());
    return *st.begin();
}
// 要素xを1つ削除. 消せたらtrueを返す.
template <class T> bool EraseOne(multiset<T>& st, const T& x) {
    auto it = st.find(x);
    if (it != st.end()) {
        st.erase(it);
        return true;
    } else
        return false;
}

/*********       C H A R       **********/
// 大文字かどうか
bool IsUpper(const char& c) {
    return isupper(c) > 0;
}
// 小文字かどうか
bool IsLower(const char& c) {
    return islower(c) > 0;
}
// 大文字へ変換
char ToUpper(const char& c) {
    string s{c}, t;
    t.resize(s.size());
    std::transform(all(s), t.begin(), ::toupper);
    return t[0];
}
// 小文字へ変換
char ToLower(const char& c) {
    string s{c}, t;
    t.resize(s.size());
    std::transform(all(s), t.begin(), ::tolower);
    return t[0];
}

/*********     S T R I N G     **********/
// 大文字へ変換
string ToUpper(const string& s) {
    string t;
    t.resize(s.size());
    std::transform(all(s), t.begin(), ::toupper);
    return t;
}
// 小文字へ変換
string ToLower(const string& s) {
    string t;
    t.resize(s.size());
    std::transform(all(s), t.begin(), ::tolower);
    return t;
}
// 回文判定, O(N). 空文字の返り値はTrue.
bool IsPalindrome(const string& S) {
    ll N = len(S);
    for (ll i = 0; 2 * i < N; i++)
        if (S[i] != S[N - i - 1]) return false;
    return true;
}

/*********       T R I O       **********/
// コンストラクタ・代入演算子
template <class T1, class T2, class T3> trio<T1, T2, T3>::trio() {
    first = T1();
    second = T2();
    third = T3();
}
template <class T1, class T2, class T3> trio<T1, T2, T3>::trio(const T1& x) : first(x), second(x), third(x) {}
template <class T1, class T2, class T3>
trio<T1, T2, T3>::trio(const T1& x, const T2& y, const T3& z) : first(x), second(y), third(z) {}
template <class T1, class T2, class T3> trio<T1, T2, T3>::trio(const trio& t) {
    first = t.first;
    second = t.second;
    third = t.third;
}
template <class T1, class T2, class T3> trio<T1, T2, T3>& trio<T1, T2, T3>::operator=(const trio& t) {
    first = t.first;
    second = t.second;
    third = t.third;
    return *this;
}
// 比較演算子 (trio)
template <class T1, class T2, class T3> bool trio<T1, T2, T3>::operator<(const trio& t) const {
    return tie(first, second, third) < tie(t.first, t.second, t.third);
}
template <class T1, class T2, class T3> bool trio<T1, T2, T3>::operator==(const trio& t) const {
    return tie(first, second, third) == tie(t.first, t.second, t.third);
}
template <class T1, class T2, class T3> bool trio<T1, T2, T3>::operator!=(const trio& t) const {
    return !(*this == t);
}
template <class T1, class T2, class T3> bool trio<T1, T2, T3>::operator>(const trio& t) const {
    return t < *this;
}
template <class T1, class T2, class T3> bool trio<T1, T2, T3>::operator<=(const trio& t) const {
    return !(*this > t);
}
template <class T1, class T2, class T3> bool trio<T1, T2, T3>::operator>=(const trio& t) const {
    return !(*this < t);
}
// 単項演算子（+ -） (trio)
template <class T1, class T2, class T3> trio<T1, T2, T3> trio<T1, T2, T3>::operator+() const {
    return *this;
}
template <class T1, class T2, class T3> trio<T1, T2, T3> trio<T1, T2, T3>::operator-() const {
    return (*this) * (-1);
}
// 入出力 (trio)
template <class T1, class T2, class T3> istream& operator>>(istream& stream, trio<T1, T2, T3>& t) {
    return stream >> t.first >> t.second >> t.third;
}
template <class T1, class T2, class T3> ostream& operator<<(ostream& stream, const trio<T1, T2, T3>& t) {
    return stream << t.first << " " << t.second << " " << t.third;
}
/* 複合代入演算子 (trio) */
template <class T1, class T2, class T3>
trio<T1, T2, T3>& operator+=(trio<T1, T2, T3>& a, const trio<T1, T2, T3> b) {
    a.fi += b.fi;
    a.se += b.se;
    a.th += b.th;
    return a;
}
template <class T1, class T2, class T3>
trio<T1, T2, T3>& operator-=(trio<T1, T2, T3>& a, const trio<T1, T2, T3> b) {
    a.fi -= b.fi;
    a.se -= b.se;
    a.th -= b.th;
    return a;
}
template <class T1, class T2, class T3>
trio<T1, T2, T3>& operator*=(trio<T1, T2, T3>& a, const trio<T1, T2, T3> b) {
    a.fi *= b.fi;
    a.se *= b.se;
    a.th *= b.th;
    return a;
}
template <class T1, class T2, class T3>
trio<T1, T2, T3>& operator/=(trio<T1, T2, T3>& a, const trio<T1, T2, T3> b) {
    a.fi /= b.fi;
    a.se /= b.se;
    a.th /= b.th;
    return a;
}
template <class T1, class T2, class T3>
trio<T1, T2, T3>& operator%=(trio<T1, T2, T3>& a, const trio<T1, T2, T3> b) {
    a.fi %= b.fi;
    a.se %= b.se;
    a.th %= b.th;
    return a;
}
template <class T1, class T2, class T3, class U>
trio<T1, T2, T3>& operator+=(trio<T1, T2, T3>& a, const U b) {
    a.fi += b;
    a.se += b;
    a.th += b;
    return a;
}
template <class T1, class T2, class T3, class U>
trio<T1, T2, T3>& operator-=(trio<T1, T2, T3>& a, const U b) {
    a.fi -= b;
    a.se -= b;
    a.th -= b;
    return a;
}
template <class T1, class T2, class T3, class U>
trio<T1, T2, T3>& operator*=(trio<T1, T2, T3>& a, const U b) {
    a.fi *= b;
    a.se *= b;
    a.th *= b;
    return a;
}
template <class T1, class T2, class T3, class U>
trio<T1, T2, T3>& operator/=(trio<T1, T2, T3>& a, const U b) {
    a.fi /= b;
    a.se /= b;
    a.th /= b;
    return a;
}
template <class T1, class T2, class T3, class U>
trio<T1, T2, T3>& operator%=(trio<T1, T2, T3>& a, const U b) {
    a.fi %= b;
    a.se %= b;
    a.th %= b;
    return a;
}
/* 算術演算子 (trio) */
template <class T1, class T2, class T3, class U> trio<T1, T2, T3> operator+(trio<T1, T2, T3> a, const U& b) {
    return a += b;
}
template <class T1, class T2, class T3, class U> trio<T1, T2, T3> operator-(trio<T1, T2, T3> a, const U& b) {
    return a -= b;
}
template <class T1, class T2, class T3, class U> trio<T1, T2, T3> operator*(trio<T1, T2, T3> a, const U& b) {
    return a *= b;
}
template <class T1, class T2, class T3, class U> trio<T1, T2, T3> operator/(trio<T1, T2, T3> a, const U& b) {
    return a /= b;
}
template <class T1, class T2, class T3, class U> trio<T1, T2, T3> operator%(trio<T1, T2, T3> a, const U& b) {
    return a %= b;
}

/*********    Q U A R T E T    **********/
// コンストラクタ・代入演算子
template <class T1, class T2, class T3, class T4> quartet<T1, T2, T3, T4>::quartet() {
    first = T1();
    second = T2();
    third = T3();
    fourth = T4();
}
template <class T1, class T2, class T3, class T4>
quartet<T1, T2, T3, T4>::quartet(const T1& x) : first(x), second(x), third(x), fourth(x) {}
template <class T1, class T2, class T3, class T4>
quartet<T1, T2, T3, T4>::quartet(const T1& x, const T2& y, const T3& z, const T4& w)
    : first(x), second(y), third(z), fourth(w) {}
template <class T1, class T2, class T3, class T4> quartet<T1, T2, T3, T4>::quartet(const quartet& t) {
    first = t.first;
    second = t.second;
    third = t.third;
    fourth = t.fourth;
}
template <class T1, class T2, class T3, class T4>
quartet<T1, T2, T3, T4>& quartet<T1, T2, T3, T4>::operator=(const quartet& t) {
    first = t.first;
    second = t.second;
    third = t.third;
    fourth = t.fourth;
    return *this;
}
// 比較演算子 (quartet)
template <class T1, class T2, class T3, class T4>
bool quartet<T1, T2, T3, T4>::operator<(const quartet& t) const {
    return tie(first, second, third, fourth) < tie(t.first, t.second, t.third, t.fourth);
}
template <class T1, class T2, class T3, class T4>
bool quartet<T1, T2, T3, T4>::operator==(const quartet& t) const {
    return tie(first, second, third, fourth) == tie(t.first, t.second, t.third, t.fourth);
}
template <class T1, class T2, class T3, class T4>
bool quartet<T1, T2, T3, T4>::operator!=(const quartet& t) const {
    return !(*this == t);
}
template <class T1, class T2, class T3, class T4>
bool quartet<T1, T2, T3, T4>::operator>(const quartet& t) const {
    return t < *this;
}
template <class T1, class T2, class T3, class T4>
bool quartet<T1, T2, T3, T4>::operator<=(const quartet& t) const {
    return !(*this > t);
}
template <class T1, class T2, class T3, class T4>
bool quartet<T1, T2, T3, T4>::operator>=(const quartet& t) const {
    return !(*this < t);
}
// 単項演算子（+ -） (quartet)
template <class T1, class T2, class T3, class T4>
quartet<T1, T2, T3, T4> quartet<T1, T2, T3, T4>::operator+() const {
    return *this;
}
template <class T1, class T2, class T3, class T4>
quartet<T1, T2, T3, T4> quartet<T1, T2, T3, T4>::operator-() const {
    return (*this) * (-1);
}
// 入出力 (quartet)
template <class T1, class T2, class T3, class T4>
istream& operator>>(istream& stream, quartet<T1, T2, T3, T4>& t) {
    return stream >> t.first >> t.second >> t.third >> t.fourth;
}
template <class T1, class T2, class T3, class T4>
ostream& operator<<(ostream& stream, const quartet<T1, T2, T3, T4>& t) {
    return stream << t.first << " " << t.second << " " << t.third << " " << t.fourth;
}
/* 複合代入演算子 (quartet) */
template <class T1, class T2, class T3, class T4>
quartet<T1, T2, T3, T4>& operator+=(quartet<T1, T2, T3, T4>& a, const quartet<T1, T2, T3, T4> b) {
    a.fi += b.fi;
    a.se += b.se;
    a.th += b.th;
    a.fo += b.fo;
    return a;
}
template <class T1, class T2, class T3, class T4>
quartet<T1, T2, T3, T4>& operator-=(quartet<T1, T2, T3, T4>& a, const quartet<T1, T2, T3, T4> b) {
    a.fi -= b.fi;
    a.se -= b.se;
    a.th -= b.th;
    a.fo -= b.fo;
    return a;
}
template <class T1, class T2, class T3, class T4>
quartet<T1, T2, T3, T4>& operator*=(quartet<T1, T2, T3, T4>& a, const quartet<T1, T2, T3, T4> b) {
    a.fi *= b.fi;
    a.se *= b.se;
    a.th *= b.th;
    a.fo *= b.fo;
    return a;
}
template <class T1, class T2, class T3, class T4>
quartet<T1, T2, T3, T4>& operator/=(quartet<T1, T2, T3, T4>& a, const quartet<T1, T2, T3, T4> b) {
    a.fi /= b.fi;
    a.se /= b.se;
    a.th /= b.th;
    a.fo /= b.fo;
    return a;
}
template <class T1, class T2, class T3, class T4>
quartet<T1, T2, T3, T4>& operator%=(quartet<T1, T2, T3, T4>& a, const quartet<T1, T2, T3, T4> b) {
    a.fi %= b.fi;
    a.se %= b.se;
    a.th %= b.th;
    a.fo %= b.fo;
    return a;
}
template <class T1, class T2, class T3, class T4, class U>
quartet<T1, T2, T3, T4>& operator+=(quartet<T1, T2, T3, T4>& a, const U b) {
    a.fi += b;
    a.se += b;
    a.th += b;
    a.fo += b;
    return a;
}
template <class T1, class T2, class T3, class T4, class U>
quartet<T1, T2, T3, T4>& operator-=(quartet<T1, T2, T3, T4>& a, const U b) {
    a.fi -= b;
    a.se -= b;
    a.th -= b;
    a.fo -= b;
    return a;
}
template <class T1, class T2, class T3, class T4, class U>
quartet<T1, T2, T3, T4>& operator*=(quartet<T1, T2, T3, T4>& a, const U b) {
    a.fi *= b;
    a.se *= b;
    a.th *= b;
    a.fo *= b;
    return a;
}
template <class T1, class T2, class T3, class T4, class U>
quartet<T1, T2, T3, T4>& operator/=(quartet<T1, T2, T3, T4>& a, const U b) {
    a.fi /= b;
    a.se /= b;
    a.th /= b;
    a.fo /= b;
    return a;
}
template <class T1, class T2, class T3, class T4, class U>
quartet<T1, T2, T3, T4>& operator%=(quartet<T1, T2, T3, T4>& a, const U b) {
    a.fi %= b;
    a.se %= b;
    a.th %= b;
    a.fo %= b;
    return a;
}
/* 算術演算子 (quartet) */
template <class T1, class T2, class T3, class T4, class U>
quartet<T1, T2, T3, T4> operator+(quartet<T1, T2, T3, T4> a, const U& b) {
    return a += b;
}
template <class T1, class T2, class T3, class T4, class U>
quartet<T1, T2, T3, T4> operator-(quartet<T1, T2, T3, T4> a, const U& b) {
    return a -= b;
}
template <class T1, class T2, class T3, class T4, class U>
quartet<T1, T2, T3, T4> operator*(quartet<T1, T2, T3, T4> a, const U& b) {
    return a *= b;
}
template <class T1, class T2, class T3, class T4, class U>
quartet<T1, T2, T3, T4> operator/(quartet<T1, T2, T3, T4> a, const U& b) {
    return a /= b;
}
template <class T1, class T2, class T3, class T4, class U>
quartet<T1, T2, T3, T4> operator%(quartet<T1, T2, T3, T4> a, const U& b) {
    return a %= b;
}

/*********     R A N D O M     **********/
// 引数なし: seedはランダム
Random::Random() {
    random_device seed_gen;
    rnd.seed(seed_gen());
}
// 引数: seed
Random::Random(ll seed) {
    rnd.seed(seed);
}
// [a, b]からランダムに整数を返す
ll Random::randint(ll a, ll b) {
    uniform_int_distribution<ll> dist(a, b);
    return dist(rnd);
}
// [a, b]からランダムに浮動小数点数を返す
double Random::randreal(double a, double b) {
    uniform_real_distribution<double> dist(a, b);
    return dist(rnd);
}
// [a, b]からランダムに文字を返す
char Random::randchar(char a, char b) {
    uniform_int_distribution<ll> dist(a, b);
    return dist(rnd);
}
// `vec`をinplaceで並び替える
template <class T> void Random::shuffle(vector<T>& vec) {
    std::shuffle(all(vec), rnd);
}

/*********       T I M E R       **********/
// 引数: タイムリミット（秒）
Timer::Timer(double time_limit_sec) {
    assert(time_limit_sec > 0);
    this->time_limit = time_limit_sec * CLOCKS_PER_SEC;
    this->start_time = clock();
}
// タイムリミットに到達したかどうか
bool Timer::is_time_up() const {
    return clock() - this->start_time >= this->time_limit;
}
// 経過時間の割合を[0.0, 1.0]で返す
double Timer::get_elapsed_ratio() const {
    double ratio = 1.0 * (clock() - this->start_time) / (this->time_limit - this->start_time);
    return min(1.0, ratio);
}

/*********       P R I N T       **********/
template <class T> void print(const T& e) {
    cout << e << '\n';
}
template <class H, class... T> void print(const H& h, const T&... t) {
    cout << h << ' ';
    print(t...);
}
template <class... T> void End(const T&... t) {
    print(t...);
    exit(0);
}

/*********       D E B U G       *********/
void dbg_(const long long& e) {
    if (e == INF)
        cout << "INF";
    else if (e == -INF)
        cout << "-INF";
    else
        cout << e;
}
template <class T> void dbg_(const T& e) {
    cout << e;
}
template <class T, class U> void dbg_(const pair<T, U>& p) {
    cout << '(';
    dbg_(p.first);
    cout << ' ';
    dbg_(p.second);
    cout << ')';
}
template <class T1, class T2, class T3> void dbg_(const trio<T1, T2, T3>& t) {
    cout << '(';
    dbg_(t.first);
    cout << ' ';
    dbg_(t.second);
    cout << ' ';
    dbg_(t.third);
    cout << ')';
}
template <class T1, class T2, class T3, class T4> void dbg_(const quartet<T1, T2, T3, T4>& t) {
    cout << '(';
    dbg_(t.first);
    cout << ' ';
    dbg_(t.second);
    cout << ' ';
    dbg_(t.third);
    cout << ' ';
    dbg_(t.fourth);
    cout << ')';
}
template <class T> void debug_(const T& e) {
    dbg_(e);
    cout << '\n';
}
template <class T> void debug_(const vector<T>& v) {
    if (v.size()) {
        auto it = v.begin();
        dbg_(*it++);
        for (; it != v.end(); ++it) {
            cout << ' ';
            dbg_(*it);
        }
    }
    cout << '\n';
}
template <class T> void debug_(const vector<vector<T>>& vv) {
    cout << '\n';
    ll cnt = 0;
    for (auto&& v : vv) {
        cout << cnt++ << ": ";
        debug_(v);
    }
}
template <class T, class U> void debug_(const map<T, U>& mp) {
    if (mp.size()) {
        auto it = mp.begin();
        dbg_(*it++);
        for (; it != mp.end(); ++it) {
            cout << ' ';
            dbg_(*it);
        }
    }
    cout << '\n';
}
template <class T, class U> void debug_(const vector<map<T, U>>& vm) {
    cout << '\n';
    ll cnt = 0;
    for (auto&& mp : vm) {
        cout << cnt++ << ": ";
        debug_(mp);
    }
}
template <class T> void debug_(const set<T>& st) {
    if (st.size()) {
        auto it = st.begin();
        dbg_(*it++);
        for (; it != st.end(); ++it) {
            cout << ' ';
            dbg_(*it);
        }
    }
    cout << '\n';
}
template <class T> void debug_(const multiset<T>& st) {
    if (st.size()) {
        auto it = st.begin();
        dbg_(*it++);
        for (; it != st.end(); ++it) {
            cout << ' ';
            dbg_(*it);
        }
    }
    cout << '\n';
}
template <class T> void debug_(const vector<set<T>>& vs) {
    cout << '\n';
    ll cnt = 0;
    for (auto&& st : vs) {
        cout << cnt++ << ": ";
        debug_(st);
    }
}
template <class T> void debug_(const vector<multiset<T>>& vs) {
    cout << '\n';
    ll cnt = 0;
    for (auto&& st : vs) {
        cout << cnt++ << ": ";
        debug_(st);
    }
}
template <class H, class... T> void debug_(const H& h, const T&... t) {
    dbg_(h);
    cout << ", ";
    debug_(t...);
}

/*********       O T H E R S       **********/
// a > b なら a に b を代入.
template <class T, class U> bool chmin(T& a, U b) {
    if (a > b) {
        a = b;
        return true;
    }
    return false;
}
// a < b なら a に b を代入.
template <class T, class U> bool chmax(T& a, U b) {
    if (a < b) {
        a = b;
        return true;
    }
    return false;
}
// a % mを返す. aは負もOK.
template <class T, class U> T Mod(const T& a, const U& m) {
    assert(m > 0);
    return (a % m + m) % m;
}
// 天井関数. xは負もOK
template <class T, class U> T Ceil(const T& x, const U& y) {
    assert(y > 0);
    return x < 0 ? x / y : (x + y - 1) / y;
}
// 床関数. xは負もOK
template <class T, class U> T Floor(const T& x, const U& y) {
    assert(y > 0);
    return -(Ceil(-x, y));
}
// 符号関数. 返り値: -1, 0, +1のいずれか
template <class T> ll Sign(const T& x) {
    return (x > T(0)) - (x < T(0));
}
// x * x <= nを満たす最大の整数`x`を返す
ll Isqrt(ll n) {
    assert(n >= 0);
    ll x = round(sqrt(n));
    while (x * x > n)
        --x;
    return x;
}
// nCr. n=60, r=30までOK.
ll Comb(ll n, ll r) {
    if (r < 0 || n < r) return 0;
    r = min(r, n - r);
    ll ret = 1;
    rep(i, r) {
        ret *= n - i;
        ret /= i + 1;
    }
    return ret;
}
// x^n. mintもOK.
template <class T> T Pow(T x, ll n) {
    assert(n >= 0);
    T ret = 1;
    while (1) {
        if (n % 2) ret *= x;
        n /= 2;
        if (!n) return ret;
        x = x * x;
    }
}
// (x^n) % mod.
template <class T> T Pow(T x, ll n, T mod) {
    assert(n >= 0 && mod > 0);
    T ret = 1;
    while (1) {
        if (n % 2) ret = (ret * x) % mod;
        n /= 2;
        if (!n) return ret;
        x = (x * x) % mod;
    }
}
// 等差級数. mintもOK.
template <class T> T Aseries(T a, T d, ll n) {
    assert(n >= 0);
    return a * n + n * (n - 1) / 2 * d;
}
// 等比級数. mintもOK.
template <class T> T Gseries(T a, T r, ll n) {
    assert(n >= 0);
    if (r == 1)
        return a * n;
    else
        return a * (1 - Pow(r, n)) / (1 - r);
}
// `b`の`i`番目（0 origin）のビットが1かどうか
bool Bit(ll b, int i) {
    assert(0 <= i && i < 64);
    return (b >> i) & 1;
}
// `b`の立っているビット数
ll Popcount(ll b) {
    return __builtin_popcountll(b);
}
// [0, n)番目のビットが立った64bit整数を返す.
ll Mask(ll n) {
    assert(0 <= n && n < 63);
    return (1LL << n) - 1LL;
}
// [m, n)番目のビットが立った64bit整数を返す.
ll Mask(ll n, ll m) {
    assert(n >= m);
    return Mask(n) ^ Mask(m);
}
// [l1, r1)と[l2, r2)の積区間を返す. 例: [3, 6) x [5, 7) = [5, 6)
pll GetOverlap(ll l1, ll r1, ll l2, ll r2) {
    return {max(l1, l2), min(r1, r2)};
}
// [l1, r1)と[l2, r2)で重なりがあるかを返す. 引数が片方でも空区間ならfalse;
bool IsOverlap(ll l1, ll r1, ll l2, ll r2) {
    return max(l1, l2) < min(r1, r2);
}
// b=trueなら"Yes\n", falseなら"No\n"を出力
void PrintYesNo(bool b) {
    b ? print("Yes") : print("No");
}
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