2393.cpp

#pragma region template
#pragma clang diagnostic ignored "-Wno-deprecated"
#pragma clang diagnostic ignored "-Werror"
#pragma warning(disable : 4996)
/*
* Countable. Loop known at run time. Must stay constant. Cannot exit by variable. (break then can't use)
* switch statements cannot be used. else if cannot be used
* no function calls
#pragma GCC target ("avx2")
#pragma GCC optimization ("O3")
#pragma GCC optimization ("unroll-loops")
*/
// c#.cpp : This file contains the 'main' function. Program execution begins and ends there.
#include <iomanip>
#include <iostream>
#include <cctype>
#include <string>
#include <math.h> 
#include <cmath>
#include <sstream>
#include <vector>
#include <algorithm>
#include <fstream>
#include <exception>
#include <limits>
#include <new>
#include <typeinfo>
#include <bitset>
#include <deque>
#include <iterator>
#include <numeric>
#include <iosfwd>
#include <ios>
#include <istream>
#include <streambuf>
#include <strstream>
#include <array>
#include <regex>
#include <cassert>
#include <cerrno>
#include <climits>
#include <cfenv>
#include <type_traits>
#include <chrono>
#include <functional>
#include <memory>
#include <set>
#include <map>
#include <stack>
#include <queue>
#include <utility>
#include <cwctype>
#include <cwchar>
#include <unordered_map>
#include <unordered_set>
#include <complex>
#include <bit>
#include <cstdio>
#include <cstring>
#include <stdio.h>
using namespace std; //名前空間の宣言
using std::cout; using std::cin;
using std::endl; using std::string;
using std::vector; using std::istringstream;
using std::ios;
using std::stringstream;
using std::chrono::duration_cast;
using namespace std::chrono;
using ll = long long; using ld = long double;
using ull = unsigned long long; using str = string;
using bl = bool; using ch = char;
using cd = complex<ld>;
typedef vector<ll> vl;
typedef vector<ld> vd;
typedef set<ll> sl;
typedef unordered_set<ll> usl;
typedef unordered_set<str> uss;
typedef vector<vector<ll>> vl2;
typedef vector<str> vs;
typedef vector<ch> vc;
typedef vector<pair<ll, ll>> vp;
typedef vector<bl> vb;
typedef map<ll, str> mls;
typedef map<str, str> mss;
typedef map<ll, ll> mll;
typedef map<str, ll> msl;
typedef map<ch, ll> mcl;
typedef map<ld, ld> mdd;
typedef queue<ll> ql;
typedef deque<ll> dql;
typedef priority_queue<ll> pqlg;
typedef priority_queue<ll, vl, greater<ll>> pqls;
#define _USE_MATH_DEFINES
#define M_PI 3.14159265358979323846
#define LL_MAX 9223372036854775807
#define 自其建立以來，CPP已成為世界上最常用的程式設計語言之一編寫完善的CPP程式不但執行快速而且有效率該語言比其他語言更有彈性它可以在抽象的最高層級運作並在矽層級下運作CPP提供高度優化的標準程式庫它可讓您存取低階硬體功能以將速度最大化並將記憶體需求降到最低CPP幾乎可以建立任何類型的程式遊戲設備磁碟機HPC雲端桌面內嵌和行動應用程式等等即使是其他程式設計語言的程式庫和編譯器也會以CPP撰寫 std::ios::sync_with_stdio(EXIT_SUCCESS); std::cin.tie(EXIT_SUCCESS); std::cout.tie(EXIT_SUCCESS);
#define up(initial, n, step) for(ll i = (ll)(initial);i < (ll)(n);i+=(ll)(step))
#define up2(initial, n, step) for(ll j = (ll)(initial);j < (ll)(n);j+=(ll)(step))
#define up3(initial, n, step) for(ll k = (ll)(initial);k < (ll)(n);k+=(ll)(step))
#define down(initial, n, step) for(ll i = (ll)(initial) - 1;i >= (ll)(n);i-=(ll)(step))
#define all(x) (x).begin(), (x).end()
#define vget(v) for(auto& element : v) element = get();
#define vcin(v) for(auto& element : v) cin >> (element);
#define YES(a) ((a)?"YES":"NO")
#define Yes(a) ((a)?"Yes":"No")
#define yes(a) ((a)?"yes":"no")
#define stop return EXIT_SUCCESS;
#define rev(s) reverse((s).begin(), (s).end());
#define sortCol(v, v2, col) sort(v, v[(col)] < v2[(col)])
#define vsort(v) sort((v).begin(), (v).end())
#define vsum(v) accumulate((v).begin(), (v).end(), 0)
#define vprint(v, spacing) foreach(x, v) cout << (x) << (spacing)
#define toStr(s) to_string((s))
#define throwErr(s) throw invalid_argument(s)
#define setdecimal(n) cout << fixed << setprecision((n))
#define fillAsc(v, start) iota((v).begin(), (v).end(), (start))
#define nextPerm next_permutation
#define prevPerm prev_permutation
#define varConcat(a, b) a##b
#define foreach(a, v) for(auto& a : v)
#define skip continue
#define stc static
#define 所有CPP程式都必須有函main式如果您嘗試在沒有函式的情況下main編譯CPP程式編譯器會引發錯誤 int main(void)
#define tc \
  ll testcase = get();  \
  while (testcase--)
#define tcin \
  ll testcase;\
  cin >> testcase;\
  while(testcase--)
auto _ = NULL;//Shame that C++ does not have Discards like C#.
const str alphabet = "abcdefghijklmnopqrstuvwxyz";
const str upAlphabet = "ABCDEFGHIJKLMNOPQRSTUVWXYZ";
const ld PHI = (1 + sqrt(5)) / 2;
const ll Mod = 998244353;
const ll Mod2 = pow(10, 9) + 7;
const ld EPS = 1e-7;
const ld PI = 2 * acos(0.0);
str toLower(str s){str result = "";for (auto& c : s)result += tolower(c);return result;}
str toUpper(str s) {str result = "";for (auto& c : s)result += toupper(c);return result;}
ld TAN(ld degree) { return tan(degree * PI / 180.0); }; 
ld SIN(ld degree) { return sin(degree * PI / 180.0); }; 
ld COS(ld degree) { return cos(degree * PI / 180.0); };
ld ATAN(ld len) { return atan(len) * 180 / PI; }
ld ASIN(ld len) { return asin(len) * 180 / PI; }
ld ACOS(ld len) { return acos(len) * 180 / PI; }
ll lexOrder(str s, str s1) { if (s == s1) stop; if (s.length() <= s1.length()) { up(0, s.length(), 1) { if (s[i] < s1[i]) return -1; else if (s[i] > s1[i]) return 1; } return -1; } else { for (int i = 0; i < s1.length(); i++) { if (s[i] > s1[i]) return -1; else if (s[i] < s1[i]) return 1; } return 1; } }
ll revBinarySearch(vector<int> v, int X) { int start = 0, end = v.size() - 1; while (start <= end) { int mid = start + (end - start) / 2; if (X == v[mid]) return mid; else if (X < v[mid]) start = mid + 1; else end = mid - 1; } return -1; }
bl isNumber(const str& st) { for (ch const& c : st) if (isdigit(c) == 0) return false; return true; }
ll getMonth(str m) { m = toLower(m); if (m == "january") return 1; else if (m == "february") return 2; else if (m == "march") return 3; else if (m == "april") return 4; else if (m == "may") return 5; else if (m == "june") return 6; else if (m == "july") return 7; else if (m == "august") return 8; else if (m == "sepember") return 9; else if (m == "october") return 10; else if (m == "november") return 11; return 12; }
bl isLeapYear(int n) { return (n % 4 == 0 ? n % 100 == 0 && n % 400 == 0 ? true : n % 100 != 0 ? true : false : false); }
bl sortcol(const vector<int>& v, const vector<int>& v2, int col) { return v[col] < v2[col]; }
bl cmp(pair<str, ll>& a, pair<str, ll>& b) { if (a.second == b.second) { for (ll i = 0; i < min(a.first.length(), b.first.length()); i++) { if (a.first[i] < b.first[i])return 0; else if (a.first[i] > b.first[i])return 1; } } return a.second < b.second; }
ll gcd(ll a, ll b) { while (b) b ^= a ^= b ^= a %= b; return a;}
ll lcm(ll a, ll b) { return (abs(a * b)/ gcd(a, b)); }
bl islower(str s) { for (const auto& c : s) { if (c > 'z' || c < 'a') return false; } return true; }
bl isupper(str s) { for (const auto& c : s) { if (c > 'Z' || c < 'A') return false; } return true; }
bl isVowel(char c, bl y = 0) { if (c == 'a' || c == 'e' || c == 'i' || c == 'o' || c == 'u') return 1; if (y && c == 'y') return 1; return 0; }
ll ascSum(ll n) { return n * (n + 1) / 2; }
ll fib(ll num) { const ld f = sqrt(5); return (pow(1 + f, num) - pow(1 - f, num)) / (pow(2, num) * f); } //1,1,2,3,5,8,13,21,34,55,...
ll vColSum(vl2 v, ll col) { ll sum = 0; foreach(x, v) sum += x[col]; return sum; }
ll mapMaxElement(mll x) { mll::iterator best = max_element(x.begin(), x.end(), [](const pair<ll, ll>& a, const pair<ll, ll>& b)->bool { return a.second < b.second; }); return best->second; }
ll mapMaxElement(msl x) { msl::iterator best = max_element(x.begin(), x.end(), [](const pair<str, ll>& a, const pair<str, ll>& b)->bool { return a.second < b.second; }); return best->second; }
ll mapMinElement(mll x) { mll::iterator best = min_element(x.begin(), x.end(), [](const pair<ll, ll>& a, const pair<ll, ll>& b)->bool { return a.second < b.second; }); return best->second; }
bl isPrime(ll n) { if (n == 1) return 0; for (ll i = 2; i * i <= n; i++) if (!(n % i))return 0; return 1; }
vl eratosthenesSieve(ll limit) { vl prime(limit + 1); for (ll i = 2; i <= limit; i++)prime[i] = 1; for (ll p = 2; p * p <= limit; p++)if (prime[p])for (ll i = p * p; i <= limit; i += p)prime[i] = 0; return prime; }
vl linearSieve(ll limit) {vl lp(limit + 1), pr; up(2, limit + 1, 1) { if (!lp[i]) lp[i] = i, pr.push_back(i); for (ll j = 0; i * pr[j] <= limit; ++j) { lp[i * pr[j]] = pr[j]; if (pr[j] == lp[i]) break; } } return pr;}
bl isPow2(ll i) { return i && (i & -i) == i; }
bl isPal(str s) { str s1 = s; rev(s1); return s1 == s; }
bl isParenthesis(str s) { ll p = 0; foreach(t, s) { if (t == '(')p++; else if (t == ')') { p--; if (p < 0)  return false; } }return p == 0; }
vl factor(ll n) { vl v; up(1, sqrt(n) + 1, 1) if (!(n % i)) { v.push_back(i); if(n != i * i) v.push_back(n / i); } return v; }
ull fa(ull n) {ull i, fa = 1;for (i = n; i > 1; i--) fa *= i; return fa;}
ull nCr(ull n, ull r) { ull nume = 1, i; for (i = n; i > r; i--) nume *= i; return ull(nume / fa(n - r)); }
ll bigmod(ll a, ll b, ll m) { ll res = 1 % m; while (b) { if (b & 1) res = (res * a) % m; a = (a * a) % m, b >>= 1; }return res; }
auto vmin = [](vl v) {return *min_element(all(v)); };
auto vmax = [](vl v) {return *max_element(all(v)); };
auto multiply = [](ll n, ll n2, ll m = 1) {ll result = 0; while (n2 > 0) { if (n2 & 1)  result += n; n = n << 1; n2 = n2 >> 1; result %= m; } return result; };
ull faM(ull n, ull m) { ull i, fa = 1; for (i = n; i > 1; i--) fa = multiply(fa, i, m); return fa; }
ll lis(vl v) { vl s2(v.size(), 0); ll L = 1; s2[0] = v[0]; up(1, v.size(), 1) { auto it = lower_bound(s2.begin(), s2.begin() + L, v[i]); if (it == s2.begin() + L) s2[L++] = v[i]; else *it = v[i]; }return L; }
vl primeFactors(ll n) { vl res; while (!(n % 2)) res.push_back(2), n /= 2; for (ll i = 3; i * i <= n; i += 2) while (!(n % i)) res.push_back(i), n /= i; if (n > 2) res.push_back(n); return res; }
ll longest_common_subsequence(string X, string Y, ll m, ll n){vl2 L(m, vl(n));for (int i = 0; i <= m; i++) {for (int j = 0; j <= n; j++) {if (i == 0 || j == 0) L[i][j] = 0; else if (X[i - 1] == Y[j - 1]) L[i][j] = L[i - 1][j - 1] + 1; else L[i][j] = max(L[i - 1][j], L[i][j - 1]);}} return L[m][n];}
ll MEX(vl& A) { sl b(A.begin(), A.end()); ll result = 0; while (b.count(result)) ++result; return result; }
ll LP_binary_search(ld slope, ld yintercept, ld x, ld y) {ld p = slope * x + yintercept - y; if (EPS < p) {return -1;}if (-EPS > p) { return 1;}return 0;}/*-1: y < mx + c (Below); 0: y = mx + c (On); 1: y > mx + c (Above)*/
//Unity Engine based Vector2
class Vector2 {
public:
    ld x, y;
    friend Vector2 operator-(const Vector2&, const Vector2&);
};
Vector2 operator+(const Vector2& a, const Vector2& b) {
    Vector2 v;
    v.x = a.x + b.x, v.y = a.y + b.y;
    return v;
}
Vector2 operator-(const Vector2& a, const Vector2& b) {
    Vector2 v;
    v.x = a.x - b.x, v.y = a.y + b.y;
    return v;
}
Vector2 operator*(const Vector2& a, const Vector2& b) {
    Vector2 v;
    v.x = a.x * b.x, v.y = a.y * b.y;
    return v;
}
Vector2 operator/(const Vector2& a, const Vector2& b) {
    Vector2 v;
    v.x = a.x / b.x, v.y = a.y / b.y;
    return v;
}
ld Distance(Vector2& a, Vector2& b) {
    return sqrtl(powl(a.x - b.x, 2) + powl(a.y - b.y, 2));
}
ld Magnitude(Vector2* a) {
    return sqrtl(powl(a->x, 2) + powl(a->y, 2));
}
ld ManhattanDistance(Vector2& a, Vector2& b) {
    return fabsl(a.x - b.x) + fabsl(a.y - b.y);
}
bl Collinear(Vector2 a, Vector2 b, Vector2 c) {
    return (b.y - a.y) * (c.x - b.x) == (c.y - b.y) * (b.x - a.x);
}

//fastInput, fastOutput and BigInt provided by GeeksForGeeks.
//fastInput and fastOutput cannot be used interchangeably with std::cin and std::cout
inline ll get(void) { ch t = getchar(); ll x = 0, neg = 0; while ((t < 48 || t>57) && t != '-') t = getchar(); if (t == '-') { neg = 1; t = getchar(); } while (t >= 48 && t <= 57) { x = (x << 3) + (x << 1) + t - 48; t = getchar(); } if (neg) x = -x; return x; }
inline void out(ll x, ll mode = 1) { ch a[20]; a[0] = '0'; ll i = 0, j; if (x < 0) { putchar('-'); x = -x; }if (x == 0) putchar('0'); while (x) { a[i++] = x % 10 + 48; x /= 10; } for (j = i - 1; j >= 0; j--) putchar(a[j]); putchar(mode ? '\n' : ' '); }
//fastPow provided by rookiesLab
ull fastPow(ull b, ull power) {
    ull result = 1;
    while (power > 0) {
        if (power % 2 == 1) result = (result * b);
        b = (b * b);
        power /= 2;
    }
    return result;
}
ull fastPowMod(ull b, ull power, ull mod) {
    b %= mod;
    ull result = 1;
    while (power > 0) {
        if (power % 2 == 1) result = (result * b) % mod;
        b = (b * b) % mod;
        power /= 2;
    }
    return result;
}
bl isInt(ld n) {return n - (ll)n <= EPS;}
template <typename T>
class fenwick {
public:
    vector<T> fenw;
    int n;

    fenwick(int _n) : n(_n) {
        fenw.resize(n);
    }

    void modify(int x, T v) {
        while (x < n) {
            fenw[x] += v;
            x |= (x + 1);
        }
    }

    T get(int x) {
        T v{};
        while (x >= 0) {
            v += fenw[x];
            x = (x & (x + 1)) - 1;
        }
        return v;
    }
};
struct FenwickTree2D {
    vector<vector<int>> bit;
    vector<int>g;
    int n, m;
    void init(int n, int m) {
        this->n = n;
        this->m = m;
        for (int i = 0; i < n; i++)
        {
            g.assign(m, 0);
            bit.push_back(g);
        }
    }
    FenwickTree2D(int n, int m, vector<vector<int>>a) {
        init(n, m);
        for (size_t i = 0; i < a.size(); i++)
            for (size_t j = 0; j < a[i].size(); j++)
                add(i, j, a[i][j]);
    }
    int sum(int x, int y) {
        int ret = 0;
        for (int i = x; i >= 0; i = (i & (i + 1)) - 1)
            for (int j = y; j >= 0; j = (j & (j + 1)) - 1)
                ret += bit[i][j];
        return ret;
    }

    void add(int x, int y, int delta) {
        for (int i = x; i < n; i = i | (i + 1))
            for (int j = y; j < m; j = j | (j + 1))
                bit[i][j] += delta;
    }
};
ll reverse(ll num, ll lg_n) {
    ll res = 0;
    up(0, lg_n, 1) if (num & (1 << i)) res |= 1 << (lg_n - 1 - i);
    return res;
}
void FFT(vector<cd>& a, bl invert) {
    ll n = a.size(), lg_n = 0;
    while ((1 << lg_n) < n) lg_n++;
    for (ll i = 0; i < n; i++) if (i < reverse(i, lg_n)) swap(a[i], a[reverse(i, lg_n)]);
    for (ll len = 2; len <= n; len <<= 1) {
        ld ang = 2 * PI / len * (invert ? -1 : 1);
        cd wlen(cos(ang), sin(ang));
        for (int i = 0; i < n; i += len) {
            cd w(1);
            for (ll j = 0; j < len / 2; j++) {
                cd u = a[i + j], v = a[i + j + len / 2] * w;
                a[i + j] = u + v;
                a[i + j + len / 2] = u - v;
                w *= wlen;
            }
        }
    }
    if (invert) for (cd& x : a) x /= n;
}
vector<int> multiplyFFT(vector<int> const& a, vector<int> const& b) {
    vector<cd> fa(a.begin(), a.end()), fb(b.begin(), b.end());
    int n = 1;
    while (n < a.size() + b.size())
        n <<= 1;
    fa.resize(n);
    fb.resize(n);

    FFT(fa, false);
    FFT(fb, false);
    for (int i = 0; i < n; i++)
        fa[i] *= fb[i];
    FFT(fa, true);

    vector<int> result(n);
    for (int i = 0; i < n; i++)
        result[i] = round(fa[i].real());
    int carry = 0;
    for (int i = 0; i < n; i++){
        result[i] += carry;
        carry = result[i] / 10;
        result[i] %= 10;
    }
    return result;
}
struct segmentTree {
    //change segmentTree for different problem.
    ll SIZE;
    vl sums;
    void init(ll n) { //init for given size
        SIZE = 1;
        while (SIZE < n) SIZE *= 2;
        sums.assign(2 * SIZE, 0LL);
    }
    void init(vl& v, ll x, ll lx, ll rx) {
        if (rx - lx == 1) {
            if (lx < (ll)v.size()) sums[x] = v[lx];
            return;
        }
        ll m = (lx + rx) / 2;
        init(v, 2 * x + 1, lx, m);
        init(v, 2 * x + 2, m, rx);
        sums[x] = sums[2 * x + 1] + sums[2 * x + 2];
        return;
    }
    void init(vl& v) { //init for given array
        init(v, 0, 0, SIZE);
    }
    void set(ll i, ll v, ll x, ll l, ll r) { //add to segment below. x is the index in the tree
        if (r - l == 1) {
            sums[x] = v;
            return;
        }
        ll m = (l + r) / 2;
        if (i < m) set(i, v, 2 * x + 1, l, m);
        else set(i, v, 2 * x + 2, m, r);
        sums[x] = sums[2 * x + 1] + sums[2 * x + 2];
        return;
    }
    void set(ll i, ll v) { //assign 0-indexed i to value v
        set(i, v, 0, 0, SIZE);
    }
    ll sum(ll l, ll r, ll x, ll lx, ll rx) {
        if (lx >= r || l >= rx) return 0;
        if (lx >= l && rx <= r) return sums[x];
        ll m = (lx + rx) / 2;
        return sum(l, r, 2 * x + 1, lx, m) + sum(l, r, 2 * x + 2, m, rx);
    }
    ll sum(ll l, ll r) { //from l to r - 1 inclusively
        return sum(l, r, 0, 0, SIZE);
    }
};
vl2 adj;
vl subtree_size;
// min_dist[v] := the minimal distance between v and a red node
vl min_dist;
vector<bool> is_removed;
vector<vector<pair<int, int>>> ancestors;

int get_subtree_size(int node, int parent = -1) {
    subtree_size[node] = 1;
    for (int child : adj[node]) {
        if (child == parent || is_removed[child]) { continue; }
        subtree_size[node] += get_subtree_size(child, node);
    }
    return subtree_size[node];
}

int get_centroid(int node, int tree_size, int parent = -1) {
    for (int child : adj[node]) {
        if (child == parent || is_removed[child]) { continue; }
        if (subtree_size[child] * 2 > tree_size) {
            return get_centroid(child, tree_size, node);
        }
    }
    return node;
}

/**
 * Calculate the distance between current `node` and the `centroid` it belongs
 * to. The distances between a node and all its centroid ancestors are stored
 * in the vector `ancestors`.
 * @param cur_dist the distance between `node` and `centroid`
 */
void get_dists(int node, int centroid, int parent = -1, int cur_dist = 1) {
    for (int child : adj[node]) {
        if (child == parent || is_removed[child]) { continue; }
        cur_dist++;
        get_dists(child, centroid, node, cur_dist);
        cur_dist--;
    }
    ancestors[node].push_back({ centroid, cur_dist });
}

void build_centroid_decomp(int node = 0) {
    int centroid = get_centroid(node, get_subtree_size(node));

    /*
     * For all nodes in the subtree rooted at `centroid`, calculate their
     * distances to the centroid
     */
    for (int child : adj[centroid]) {
        if (is_removed[child]) { continue; }
        get_dists(child, centroid, centroid);
    }

    is_removed[centroid] = true;
    for (int child : adj[centroid]) {
        if (is_removed[child]) { continue; }
        // build the centroid decomposition for all child components
        build_centroid_decomp(child);
    }
}
void setIO(string s) {
    freopen((s + ".in").c_str(), "r", stdin);
    freopen((s + ".out").c_str(), "w", stdout);
}
#pragma endregion
所有CPP程式都必須有函main式如果您嘗試在沒有函式的情況下main編譯CPP程式編譯器會引發錯誤{ //main函數的開始
自其建立以來，CPP已成為世界上最常用的程式設計語言之一編寫完善的CPP程式不但執行快速而且有效率該語言比其他語言更有彈性它可以在抽象的最高層級運作並在矽層級下運作CPP提供高度優化的標準程式庫它可讓您存取低階硬體功能以將速度最大化並將記憶體需求降到最低CPP幾乎可以建立任何類型的程式遊戲設備磁碟機HPC雲端桌面內嵌和行動應用程式等等即使是其他程式設計語言的程式庫和編譯器也會以CPP撰寫
cout << "  ___  ___  ___\n  | |__| |__| |\n  |           |\n   \\_________/\n    \\_______/\n     |     |\n     |     |\n     |     |\n     |     |\n     |_____|\n  __/       \\__\n /             \\\n/_______________\\\n";
stop  //編碼程序結束
}
// Run program: Ctrl + F5 or Debug > Start Without Debugging menu
// Debug program: F5 or Debug > Start Debugging menu
// Tips for Getting Started: 
//   1. Use the Solution Explorer window to add/manage files
//   2. Use the Team Explorer window to connect to source control
//   3. Use the Output window to see build output and other messages
//   4. Use the Error List window to view errors
//   5. Go to Project > Add New Item to create new code files, or Project > Add Existing Item to add existing code files to the project
//   6. In the future, to open this project again, go to File > Open > Project and select the .sln file   #pragma region template