#pragma region Macros
 
#pragma GCC optimize("O3,unroll-loops")
#pragma GCC target("sse,sse2,sse3,ssse3,sse4,fma,mmx,abm,bmi,bmi2,popcnt,lzcnt")
#pragma GCC target("avx2") // CF, CodeChef, HOJ ではコメントアウト
 
#include <bits/extc++.h>
// #include <atcoder/all>
// using namespace atcoder;
using namespace std;
using namespace __gnu_pbds;
 
// #include <boost/multiprecision/cpp_dec_float.hpp>
// #include <boost/multiprecision/cpp_int.hpp>
// namespace mp = boost::multiprecision;
// using Bint = mp::cpp_int;
// using Bdouble = mp::number<mp::cpp_dec_float<256>>;
// Bdouble Beps = 0.00000000000000000000000000000001; // 1e-32
// const bool equals(Bdouble a, Bdouble b) { return mp::fabs(a - b) < Beps; }

#define pb emplace_back
#define int ll
#define endl '\n'
 
// #define sqrt __builtin_sqrtl
// #define cbrt __builtin_cbrtl
// #define hypot __builtin_hypotl
 
using ll = long long;
using ld = long double;
const ld PI = acosl(-1);
const int INF = 1 << 30;
const ll INFL = 1LL << 61;
const int MOD = 998244353;
// const int MOD = 1000000007;

const ld EPS = 1e-10;
const bool equals(ld a, ld b) { return fabs((a) - (b)) < EPS; }
 
const vector<int> dx = {0, 1, 0, -1, 1, 1, -1, -1, 0}; // → ↓ ← ↑ ↘ ↙ ↖ ↗ 自
const vector<int> dy = {1, 0, -1, 0, 1, -1, -1, 1, 0};
 
#define EC int
struct Edge {
    int from, to;
    EC cost;
    Edge() {}
    // Edge() : from(-1), to(-1), cost(-1) {}
    Edge(int to, EC cost) : to(to), cost(cost) {}
    Edge(int from, int to, EC cost) : from(from), to(to), cost(cost) {}
    bool operator ==(const Edge &e) {
        return this->from == e.from && this->to == e.to && this->cost == e.cost;
    }
    bool operator !=(const Edge &e) {
        return this->from != e.from or this->to != e.to or this->cost != e.cost;
    }
    bool operator <(const Edge &e) { return this->cost < e.cost; }
    bool operator >(const Edge &e) { return this->cost > e.cost; }
};
 
chrono::system_clock::time_point start;
__attribute__((constructor))
void constructor() {
    ios::sync_with_stdio(false);
    cin.tie(nullptr);
    cout << fixed << setprecision(10);
    start = chrono::system_clock::now();
}
 
random_device seed_gen;
mt19937_64 rng(seed_gen());
uniform_int_distribution<int> dist_x(0, 1e9);
struct RNG {
    unsigned Int() {
        return dist_x(rng);
    }
    unsigned Int(unsigned l, unsigned r) {
        return dist_x(rng) % (r - l + 1) + l;
    }
    ld Double() {
        return ld(dist_x(rng)) / 1e9;
    }
} rnd;

namespace bit_function {
    using i64 = ll;
    // using i64 = uint64_t;
    // bit演算, x==0の場合は例外処理した方がよさそう. 区間は [l, r)
    i64 lrmask(int l, int r) { return (1LL << r) - (1LL << l); }
    i64 sub_bit(i64 x, int l, int r) { i64 b = x & ((1LL << r) - (1LL << l)); return b >> l; } // r溢れ可
    i64 bit_width(i64 x) { return 64 - __builtin_clzll(x) + (x == 0); }
    
    i64 popcount(i64 x) { return __builtin_popcountll(x); }
    i64 popcount(i64 x, int l, int r) { return __builtin_popcountll(sub_bit(x, l, r)); }
    i64 unpopcount(i64 x) { return bit_width(x) - __builtin_popcountll(x); } // 最上位bitより下のみ
    i64 unpopcount(i64 x, int l, int r) { return r - l - __builtin_popcountll(sub_bit(x, l, r)); } // 最上位bitより上も含まれうる
    bool is_pow2(i64 x) { return __builtin_popcountll(x) == 1; } // xが負のときは常にfalse
    bool is_pow4(i64 x) { return __builtin_popcountll(x) == 1 && __builtin_ctz(x) % 2 == 0; }
    //bool is_pow4(ll x) { return __builtin_popcountll(x) == 1 && (x&0x55555555); }
    
    int top_bit(i64 x) { return 63 - __builtin_clzll(x);} // 2^kの位 (x > 0)
    int bot_bit(i64 x) { return __builtin_ctzll(x);} // 2^kの位 (x > 0)
    int next_bit(i64 x, int k) { // upper_bound
        x >>= (k + 1);
        int pos = k + 1;
        while (x > 0) {
            if (x & 1) return pos;
            x >>= 1;
            pos++;
        }
        return -1;
    }
    int prev_bit(i64 x, int k) {
        // k = min(k, bit_width(x)); ?
        int pos = 0;
        while (x > 0 && pos < k) {
            if (x & 1) {
                if (pos < k) return pos;
            }
            x >>= 1;
            pos++;
        }
        return -1;
    }
    int kth_bit(i64 x, int k) { // kは1-indexed
        int pos = 0, cnt = 0;
        while (x > 0) {
            if (x & 1) {
                cnt++;
                if (cnt == k) return pos;
            }
            x >>= 1;
            pos++;
        }
        return -1;
    }
    i64 msb(i64 x) { if (x == 0) return 0; return 1LL << (63 - __builtin_clzll(x)); } // mask
    i64 lsb(i64 x) { return (x & -x); } // mask
    
    int countl_zero(i64 x) { return __builtin_clzll(x); }
    int countl_one(i64 x) { // countl_oneと定義が異なるので注意
        i64 ret = 0, k = 63 - __builtin_clzll(x);
        while (k != -1 && (x & (1LL << k))) { k--; ret++; }
        return ret;
    }
    int countr_zero(i64 x) { return __builtin_ctzll(x); } // x=0のとき64
    int countr_one(i64 x) { int ret = 0; while (x & 1) { x >>= 1; ret++; } return ret; }
    // int countr_one(ll x) { return __builtin_popcount(x ^ (x & -~x));

    i64 l_one(i64 x) { // 最上位で連なってる1のmask
        if (x == 0) return 0;
        i64 ret = 0, k = 63 - __builtin_clzll(x);
        while (k != -1 && (x & (1LL << k))) { ret += 1LL << k; k--; }
        return ret;
    }
    
    int floor_log2(i64 x) { return 63 - __builtin_clzll(x); } // top_bit
    int ceil_log2(i64 x) { return 64 - __builtin_clzll(x - 1); }
    i64 bit_floor(i64 x) { if (x == 0) return 0; return 1LL << (63 - __builtin_clzll(x)); } // msb
    i64 bit_ceil(i64 x) { if (x == 0) return 0; return 1LL << (64 - __builtin_clzll(x - 1)); }
    
    i64 rotl(i64 x, int k) { // 有効bit内でrotate. オーバーフロー注意
        i64 w = bit_width(x); k %= w;
        return ((x << k) | (x >> (w - k))) & ((1LL << w) - 1);
    }
    // i64 rotl(i64 x, i64 l, i64 m, i64 r) {}
    i64 rotr(i64 x, int k) {
        i64 w = bit_width(x); k %= w;
        return ((x >> k) | (x << (w - k))) & ((1LL << w) - 1);
    }
    // i64 rotr(i64 x, i64 l, i64 m, i64 r) {}
    i64 bit_reverse(i64 x) { // 有効bit内で左右反転
        i64 r = 0, w = bit_width(x);
        for (i64 i = 0; i < w; i++) r |= ((x >> i) & 1) << (w - i - 1);
        return r;
    }
    // i64 bit_reverse(i64 x, int l, int r) {}
    
    bool is_palindrome(i64 x) { return x == bit_reverse(x); }
    bool is_palindrome(i64 x, int l, int r) { i64 b = sub_bit(x, l, r); return b == bit_reverse(b); }
    i64 concat(i64 a, i64 b) { return (a << bit_width(b)) | b; } // オーバーフロー注意
    i64 erase(i64 x, int l, int r) { return x >> r << l | x & ((1LL << l) - 1); } // [l, r) をカット
    
    i64 hamming(i64 a, i64 b) { return __builtin_popcountll(a ^ b); }
    i64 hamming(i64 a, i64 b, int l, int r) { return __builtin_popcountll(sub_bit(a, l, r) ^ sub_bit(b, l, r)); }
    i64 compcount(i64 x) { return (__builtin_popcountll(x ^ (x >> 1)) + (x & 1)) / 2; }
    i64 compcount2(i64 x) { return compcount(x & (x >> 1)); } // 長さ2以上の連結成分の個数
    i64 adjacount(i64 x) { return __builtin_popcountll(x & (x >> 1)); } // 隣接する1のペアの個数
    
    i64 next_combination(i64 x) {
        i64 t = x | (x - 1); return (t + 1) | (((~t & -~t) - 1) >> (__builtin_ctzll(x) + 1));
    }
} using namespace bit_function;

namespace util_function {
    namespace Std = std;
    __int128_t POW(__int128_t x, int n) {
        __int128_t ret = 1;
        assert(n >= 0);
        if (x == 1 or n == 0) ret = 1;
        else if (x == -1 && n % 2 == 0) ret = 1; 
        else if (x == -1) ret = -1; 
        else if (n % 2 == 0) {
            // assert(x < INFL);
            ret = POW(x * x, n / 2);
        } else {
            // assert(x < INFL);
            ret = x * POW(x, n - 1);
        }
        return ret;
    }
    int per(int x, int y) { // x = qy + r (0 <= r < y) を満たすq
        assert(y != 0);
        if (x >= 0 && y > 0) return x / y;
        if (x >= 0 && y < 0) return x / y - (x % y < 0);
        if (x < 0 && y < 0) return x / y + (x % y < 0);
        return x / y - (x % y < 0); //  (x < 0 && y > 0) 
    }
    int mod(int x, int y) { // x = qy + r (0 <= r < y) を満たすr
        assert(y != 0);
        return x - y * per(x, y);
    } // https://yukicoder.me/problems/no/2781
    int floor(int x, int y) { // (ld)x / y 以下の最大の整数
        assert(y != 0);
        if (y < 0) x = -x, y = -y;
        return x >= 0 ? x / y : (x + 1) / y - 1;
    }
    int ceil(int x, int y) { // (ld)x / y 以上の最小の整数
        assert(y != 0);
        if (y < 0) x = -x, y = -y;
        return x > 0 ? (x - 1) / y + 1 : x / y;
    }
    int round(int x, int y) { // (ld)x / y を小数第1位について四捨五入
        assert(y != 0);
        return (x * 2 + y) / (y * 2);
    }
    int round(int x, int y, int k) { // (ld)x / y を10^kの位に関して四捨五入
        assert(y != 0 && k >= 0);
        if (k == 0) return (x * 2 + y) / (y * 2);
        x /= y * POW(10, k - 1);
        if (x % 10 >= 5) return (x + 10 - x % 10) * POW(10, k - 1);
        return x * POW(10, k - 1);
    }
    int round2(int x, int y) { // 五捨五超入 // 未verify
        assert(y != 0);
        if (y < 0) y = -y, x = -x;
        int z = x / y;
        if ((z * 2 + 1) * y <= y * 2) z++;
        return z;
    }
    ld round(ld x, int k) { // xを10^kの位に関して四捨五入.
        // x += EPS;
        ld d = pow(10, -k);
        return Std::round(x * d) / d;
    }
    ld floor(ld x, int k) { // xを10^kの位に関してflooring
        // x += EPS;
        ld d = pow(10, -k);
        return Std::floor(x * d) / d; // 未verify
    }
    ld ceil(ld x, int k) { // xを10^kの位に関してceiling
        // x -= EPS;
        ld d = pow(10, -k);
        return Std::ceil(x * d) / d; // 未verify
    }
    // int kth(int x, int y, int k) { // x / yの10^kの位の桁
    // }
    int floor(ld x, ld y) { // 誤差対策TODO
        assert(!equals(y, 0));
        return Std::floor(x / y);
        // floor(x) = ceil(x - 1) という話も
    }
    int ceil(ld x, ld y) { // 誤差対策TODO // ceil(p/q) = -floor(-(p/q))らしい
        assert(!equals(y, 0));
        return Std::ceil(x / y);
        // ceil(x) = floor(x + 1)
    }
    int perl(ld x, ld y) { // x = qy + r (0 <= r < y, qは整数) を満たす q
        // 未verify. 誤差対策TODO. EPS外してもいいかも。
        assert(!equals(y, 0));
        if (x >= 0 && y > 0) return Std::floor(x / y)+EPS;
        if (x >= 0 && y < 0) return -Std::floor(x / fabs(y));
        if (x < 0 && y < 0) return Std::floor(x / y) + (x - Std::floor(x/y)*y < -EPS);
        return Std::floor(x / y) - (x - Std::floor(x/y)*y < -EPS); //  (x < 0 && y > 0) 
    }
    ld modl(ld x, ld y) { // x = qy + r (0 <= r < y, qは整数) を満たす r
        // 未verify. 誤差対策TODO. -0.0が返りうる。
        assert(!equals(y, 0));
        if (x >= 0) return x - fabs(y)*fabs(per(x, y));
        return x - fabs(y)*floor(x, fabs(y));
    }
    int seisuu(ld x) { return (int)x; } // 整数部分. 誤差対策TODO
    int modf(ld x) {
        if (x < 0) return ceill(x);
        else return floorl(x);
    }
    // 正なら+EPS, 負なら-EPSしてから、文字列に直して小数点以下を捨てる?
    int seisuu(int x, int y) {
        assert(y != 0);
        return x / y;
    }
    int seisuu(ld x, ld y) { // 誤差対策TODO
        assert(!equals(y, 0));
        return (int)(x / y);
    }

    int floor_log(int base, int x) {
        assert(base >= 2);
        int ret = 0, now = 1;
        while (now <= x) {
            now *= base;
            if (now <= x) ret++;
        }
        return ret;
    }
    int ceil_log(int base, int x) {
        assert(base >= 2);
        int ret = 0, now = 1;
        while (now < x) {
            now *= base;
            ret++;
        }
        return ret;
    }

    template <class T> pair<T, T> max(const pair<T, T> &a, const pair<T, T> &b) {
        if (a.first > b.first or a.first == b.first && a.second > b.second) return a;
        return b;
    }
    template <class T> pair<T, T> min(const pair<T, T> &a, const pair<T, T> &b) {
        if (a.first < b.first or a.first == b.first && a.second < b.second) return a;
        return b;
    }
    
    template <class T> bool chmax(T &a, const T &b) {
        if (a < b) { a = b; return true; } return false;
    }
    template <class T> bool chmin(T &a, const T &b) {
        if (a > b) { a = b; return true; } return false;
    }
    template <class T> bool chmax(pair<T, T> &a, const pair<T, T> &b) {
        if (a.first < b.first or a.first == b.first && a.second < b.second) { a = b; return true; }
        return false;
    }
    template <class T> bool chmin(pair<T, T> &a, const pair<T, T> &b) {
        if (a.first > b.first or a.first == b.first && a.second > b.second) { a = b; return true; }
        return false;
    }
    template <class T> T mid(T a, T b, T c) { // 誤差対策TODO
        return a + b + c - Std::max({a, b, c}) - Std::min({a, b, c});
    }
    template <typename T, typename... Args>
    void Sort(T& a, T& b, T& c, Args&... args) {
        vector<T> vec = {a, b, c, args...};
        sort(vec.begin(), vec.end());
        auto it = vec.begin();
        a = *it++; b = *it++; c = *it++;
        int dummy[] = { (args = *it++, 0)... };
        static_cast<void>(dummy);
    }
    template <typename T, typename... Args>
    void Sortr(T& a, T& b, T& c, Args&... args) {
        vector<T> vec = {a, b, c, args...};
        sort(vec.rbegin(), vec.rend());
        auto it = vec.begin();
        a = *it++; b = *it++; c = *it++;
        int dummy[] = { (args = *it++, 0)... };
        static_cast<void>(dummy);
    }
    template <class T>
    void sort(vector<T> &A, vector<T> &B) {
        vector<pair<T, T>> P(A.size());
        for (int i = 0; i < A.size(); i++) P[i] = {A[i], B[i]};
        sort(P.begin(), P.end());
        for (int i = 0; i < A.size(); i++) A[i] = P[i].first, B[i] = P[i].second;
    }

    istream &operator >>(istream &is, __int128_t& x) {
        string S; is >> S;
        __int128_t ret = 0;
        int f = 1;
        if (S[0] == '-') f = -1; 
        for (int i = 0; i < S.length(); i++)
            if ('0' <= S[i] && S[i] <= '9')
                ret = ret * 10 + S[i] - '0';
        x = ret * f;
        return (is);
    }
    ostream &operator <<(ostream &os, __int128_t x) {
        ostream::sentry s(os);
        if (s) {
            __uint128_t tmp = x < 0 ? -x : x;
            char buffer[128]; char *d = end(buffer);
            do {
                --d; *d = "0123456789"[tmp % 10]; tmp /= 10;
            } while (tmp != 0);
            if (x < 0) { --d; *d = '-'; }
            int len = end(buffer) - d;
            if (os.rdbuf()->sputn(d, len) != len) os.setstate(ios_base::badbit);
        }
        return os;
    }
    
    __int128_t sto128(const string &S) {
        __int128_t ret = 0; int f = 1;
        if (S[0] == '-') f = -1; 
        for (int i = 0; i < S.length(); i++)
            if ('0' <= S[i] && S[i] <= '9') ret = ret * 10 + S[i] - '0';
        return ret * f;
    }
    __int128_t gcd(__int128_t a, __int128_t b) { return b ? gcd(b, a % b) : a; }
    __int128_t lcm(__int128_t a, __int128_t b) {
        return a / gcd(a, b) * b;
        // lcmが__int128_tに収まる必要あり
    }
    
    string to_string(double x, int k) { // 小数第k+1を四捨五入して小数第k位までを出力
    // 切り捨てがほしい場合は to_string(x, k+1) として pop_back() すればよい?
        ostringstream os;
        os << fixed << setprecision(k) << x;
        return os.str();
    }
    string to_string(__int128_t x) {
        string ret = "";
        if (x < 0) { ret += "-"; x *= -1; }
        while (x) { ret += (char)('0' + x % 10); x /= 10; }
        reverse(ret.begin(), ret.end());
        return ret;
    }
    string to_string(char c) { string s = ""; s += c; return s; }
} using namespace util_function;

struct custom_hash {
    static uint64_t splitmix64(uint64_t x) {
        x += 0x9e3779b97f4a7c15;
        x = (x ^ (x >> 30)) * 0xbf58476d1ce4e5b9;
        x = (x ^ (x >> 27)) * 0x94d049bb133111eb;
        return x ^ (x >> 31);
    }
 
    size_t operator()(uint64_t x) const {
        static const uint64_t FIXED_RANDOM = chrono::steady_clock::now().time_since_epoch().count();
        return splitmix64(x + FIXED_RANDOM);
    }
};

template<class T> size_t HashCombine(const size_t seed,const T &v) {
    return seed^(hash<T>()(v)+0x9e3779b9+(seed<<6)+(seed>>2));
}
template<class T,class S> struct hash<pair<T,S>>{
    size_t operator()(const pair<T,S> &keyval) const noexcept {
        return HashCombine(hash<T>()(keyval.first), keyval.second);
    }
};
template<class T> struct hash<vector<T>>{
    size_t operator()(const vector<T> &keyval) const noexcept {
        size_t s=0;
        for (auto&& v: keyval) s=HashCombine(s,v);
        return s;
    }
};
template<int N> struct HashTupleCore{
    template<class Tuple> size_t operator()(const Tuple &keyval) const noexcept{
        size_t s=HashTupleCore<N-1>()(keyval);
        return HashCombine(s,get<N-1>(keyval));
    }
};
template <> struct HashTupleCore<0>{
    template<class Tuple> size_t operator()(const Tuple &keyval) const noexcept{ return 0; }
};
template<class... Args> struct hash<tuple<Args...>>{
    size_t operator()(const tuple<Args...> &keyval) const noexcept {
        return HashTupleCore<tuple_size<tuple<Args...>>::value>()(keyval);
    }
};

template<typename T>
class Compress {
public:
    int sz = 0;
    vector<T> uniqV;

    Compress() {}
    
    template<typename... Vecs>
    Compress(const Vecs&... vecs) {
        (uniqV.insert(uniqV.end(), vecs.begin(), vecs.end()), ...);
        sort(uniqV.begin(), uniqV.end());
        uniqV.erase(unique(uniqV.begin(), uniqV.end()), uniqV.end());
        sz = uniqV.size();
    }

    vector<int> zip(const vector<T> &V) {
        vector<int> ret(V.size());
        for (int i = 0; i < V.size(); i++) {
            ret[i] = encode(V[i]);
        }
        return ret;
    }

    vector<T> unzip(const vector<int> &V) {
        vector<T> ret(V.size());
        for (int i = 0; i < V.size(); i++) {
            ret[i] = decode(V[i]);
        }
        return ret;
    }

    int size() { return sz; }

    int encode(T x) {
        auto it = lower_bound(uniqV.begin(), uniqV.end(), x);
        return it - uniqV.begin();
    }

    T decode(int x) {
        if (x < 0 or x >= uniqV.size()) return -1; // xが範囲外の場合
        return uniqV[x];
    }
};
 
class UnionFind {
public:
	UnionFind() = default;
    UnionFind(int N) : par(N), sz(N, 1) {
        iota(par.begin(), par.end(), 0);
    }
	int root(int x) {
		if (par[x] == x) return x;
		return (par[x] = root(par[x]));
	}
	bool unite(int x, int y) {
		int rx = root(x);
		int ry = root(y);
        if (rx == ry) return false;
		if (sz[rx] < sz[ry]) swap(rx, ry);
		sz[rx] += sz[ry];
		par[ry] = rx;
        return true;
	}
	bool issame(int x, int y) { return (root(x) == root(y)); }
	int size(int x) { return sz[root(x)]; }
    vector<vector<int>> groups(int N) {
        vector<vector<int>> G(N);
        for (int x = 0; x < N; x++) {
            G[root(x)].push_back(x);
        }
		G.erase( remove_if(G.begin(), G.end(),
            [&](const vector<int>& V) { return V.empty(); }), G.end());
        return G;
    }
private:
	vector<int> par, sz;
};
 
template<typename T> struct BIT {
    int N;             // 要素数
    vector<T> bit[2];  // データの格納先
    BIT(int N_, int x = 0) {
        N = N_ + 1;
        bit[0].assign(N, 0); bit[1].assign(N, 0);
        if (x != 0) {
            for (int i = 0; i < N; i++) add(i, x);
        }
    }
    BIT(const vector<T> &A) {
        N = A.size() + 1;
        bit[0].assign(N, 0); bit[1].assign(N, 0);
        for (int i = 0; i < (int)A.size(); i++) add(i, A[i]);
    }
    void add_sub(int p, int i, T x) {
        while (i < N) { bit[p][i] += x; i += (i & -i); }
    }
    void add(int l, int r, T x) {
        add_sub(0, l + 1, -x * l); add_sub(0, r + 1, x * r);
        add_sub(1, l + 1, x); add_sub(1, r + 1, -x);
    }
    void add(int i, T x) { add(i, i + 1, x); }
    T sum_sub(int p, int i) {
        T ret = 0;
        while (i > 0) { ret += bit[p][i]; i -= (i & -i); }
        return ret;
    }
    T sum(int i) { return sum_sub(0, i) + sum_sub(1, i) * i; }
    T sum(int l, int r) { return sum(r) - sum(l); }
    T get(int i) { return sum(i, i + 1); }
    void set(int i, T x) { T s = get(i); add(i, -s + x); }
};
 
template<int mod> class Modint {
public:
    int val = 0;
    Modint(int x = 0) { while (x < 0) x += mod; val = x % mod; }
    Modint(const Modint &r) { val = r.val; }
 
    Modint operator -() { return Modint(-val); } // 単項
    Modint operator +(const Modint &r) { return Modint(*this) += r; }
    Modint operator +(const int &q) { Modint r(q); return Modint(*this) += r; }
    Modint operator -(const Modint &r) { return Modint(*this) -= r; }
    Modint operator -(const int &q) { Modint r(q); return Modint(*this) -= r; }
    Modint operator *(const Modint &r) { return Modint(*this) *= r; }
    Modint operator *(const int &q) { Modint r(q); return Modint(*this) *= r; }
    Modint operator /(const Modint &r) { return Modint(*this) /= r; }
    Modint operator /(const int &q) { Modint r(q); return Modint(*this) /= r; }
    
    Modint& operator ++() { val++; if (val >= mod) val -= mod; return *this; } // 前置
    Modint operator ++(signed) { ++*this; return *this; } // 後置
    Modint& operator --() { val--; if (val < 0) val += mod; return *this; }
    Modint operator --(signed) { --*this; return *this; }
    Modint &operator +=(const Modint &r) { val += r.val; if (val >= mod) val -= mod; return *this; }
    Modint &operator +=(const int &q) { Modint r(q); val += r.val; if (val >= mod) val -= mod; return *this; }
    Modint &operator -=(const Modint &r) { if (val < r.val) val += mod; val -= r.val; return *this; }
    Modint &operator -=(const int &q) { Modint r(q);  if (val < r.val) val += mod; val -= r.val; return *this; }
    Modint &operator *=(const Modint &r) { val = val * r.val % mod; return *this; }
    Modint &operator *=(const int &q) { Modint r(q); val = val * r.val % mod; return *this; }
    Modint &operator /=(const Modint &r) {
        int a = r.val, b = mod, u = 1, v = 0;
        while (b) {int t = a / b; a -= t * b; swap(a, b); u -= t * v; swap(u, v);}
        val = val * u % mod; if (val < 0) val += mod;
        return *this;
    }
    Modint &operator /=(const int &q) {
        Modint r(q); int a = r.val, b = mod, u = 1, v = 0;
        while (b) {int t = a / b; a -= t * b; swap(a, b); u -= t * v; swap(u, v);}
        val = val * u % mod; if (val < 0) val += mod;
        return *this;
    }
    bool operator ==(const Modint& r) { return this -> val == r.val; }
    bool operator <(const Modint& r) { return this -> val < r.val; }
    bool operator >(const Modint& r) { return this -> val > r.val; }
    bool operator !=(const Modint& r) { return this -> val != r.val; }

    friend istream &operator >>(istream &is, Modint& x) {
        int t; is >> t; x = t; return (is);
    }
    friend ostream &operator <<(ostream &os, const Modint& x) {
        return os << x.val;
    }
};
using mint = Modint<MOD>;
 
mint modpow(const mint &x, int n) {
    if (n < 0) return (mint)1 / modpow(x, -n); // 未verify
    assert(n >= 0);
    if (n == 0) return 1;
    mint t = modpow(x, n / 2);
    t = t * t;
    if (n & 1) t = t * x;
    return t;
}
int modpow(__int128_t x, int n, int mod) {
    if (n == 0 && mod == 1) return 0;
    assert(n >= 0 && mod > 0); // TODO: n <= -1
    __int128_t ret = 1;
    while (n > 0) {
        if (n % 2 == 1) ret = ret * x % mod;
        x = x * x % mod;
        n /= 2;
    }
    return ret;
}
// int modinv(__int128_t x, int mod) { // 
//     assert(mod > 0);
//     // assert(x > 0);
//     if (x == 1 or x == 0) return 1;
//     return mod - modinv(mod % x, mod) * (mod / x) % mod;
// }

vector<mint> _fac, _finv, _inv;
void COMinit(int N) {
    _fac.resize(N + 1); _finv.resize(N + 1);  _inv.resize(N + 1);
    _fac[0] = _fac[1] = 1; _finv[0] = _finv[1] = 1; _inv[1] = 1;
    for (int i = 2; i <= N; i++) {
        _fac[i] = _fac[i-1] * mint(i);
        _inv[i] = -_inv[MOD % i] * mint(MOD / i);
        _finv[i] = _finv[i - 1] * _inv[i];
    }
}
 
mint FAC(int N) {
    if (N < 0) return 0; return _fac[N];
}
mint FACinv(int N) {
    if (N < 0) return 0; return _finv[N];
}
mint COM(int N, int K) {
    if (N < K) return 0; if (N < 0 or K < 0) return 0;
    return _fac[N] * _finv[K] * _finv[N - K];
}
mint COMinv(int N, int K) {
    if (N < K) return 0; if (N < 0 or K < 0) return 0;
    return _finv[N] * _fac[K] * _fac[N - K];
}
mint MCOM(const vector<int> &V) {
    int N = 0;
    for (int i = 0; i < V.size(); i++) N += V[i];
    mint ret = _fac[N];
    for (int i = 0; i < V.size(); i++) ret *= _finv[V[i]];
    return ret;
}
mint PERM(int N, int K) {
    if (N < K) return 0; if (N < 0 or K < 0) return 0;
    return _fac[N] *  _finv[N - K];
}
mint NHK(int N, int K) { // initのサイズに注意
    if (N == 0 && K == 0)  return 1;
    return COM(N + K - 1, K);
}
 
#pragma endregion

struct Point3d {
    double x, y, z;

    Point3d() {}
    Point3d(double x, double y, double z) : x(x), y(y), z(z) {}

    Point3d operator+(const Point3d &p) { return Point3d(x+p.x, y+p.y, z+p.z); }
    Point3d operator-(const Point3d &p) { return Point3d(x-p.x, y-p.y, z-p.z); }
    Point3d operator*(const double &k) { return Point3d(x*k, y*k, z*k); }
    Point3d operator/(const double &k) { return Point3d(x/k, y/k, z/k); }

    friend istream& operator>>(istream &is, Point3d &p) {
		is >> p.x >> p.y >> p.z; return is;
	}
	friend ostream& operator<<(ostream& os, Point3d &p) {
		os << p.x << " " << p.y << " " << p.z; return os;
	}

    bool operator==(const Point3d &p) const {
        return equals(x, p.x) && equals(y, p.y) && equals(z, p.z);
    }
    bool operator<(const Point3d &p) const {
        if (!equals(x, p.x)) return x < p.x;
        if (!equals(y, p.y)) return y < p.y;
        if (!equals(z, p.z)) return z < p.z;
        return false;
    }
};
using Vector3d = Point3d;

int sign(double x) { return x < -EPS ? -1 : x > EPS; } // -1(負)/0/1(正)

double norm(Point3d p) { return p.x*p.x + p.y*p.y + p.z*p.z; }
double abs(Point3d p) { return sqrt(norm(p)); }
double dist(Point3d a, Point3d b) { return sqrt(norm(a - b)); }
double dot(Point3d a, Point3d b) { return a.x*b.x + a.y*b.y + a.z*b.z; }
Vector3d cross(Point3d a, Point3d b) {
    return Vector3d(a.y*b.z - a.z*b.y, a.z*b.x - a.x*b.z, a.x*b.y - a.y*b.x);
} // https://yukicoder.me/problems/no/2765
Vector3d normalize(Vector3d v) {  // 長さを1に正規化(単位ベクトル)
    assert(abs(v) > EPS);
    return v / abs(v);
}

double rad_to_deg(double rad) { return (rad * 180. / PI); }
double deg_to_rad(double deg) { return (deg * PI / 180.); }
double normalize(double rad) { // to [-PI, PI)
	double ret = fmod(rad + PI, PI*2.);
	if (ret < 0) ret += PI;
	else ret -= PI;
	return ret;
} // maroon's library, Contest 2: ptKU Contest 1, ptTZ Summer 2022 Day 4

double angle(Point3d a, Point3d b) {
    double costheta = dot(a, b) / norm(a) / norm(b);
    return acos(max<double>(-1., min<double>(1., costheta)));
}
double small_angle(Point3d a, Point3d b) {
    return acos(min(fabs(dot(a, b)) / abs(a) / abs(b), (double)1.));
}
bool is_lattice(Point3d p) { // 未
    return equals(abs(p.x - round(p.x)), 0.) &&  equals(abs(p.y - round(p.y)), 0.) &&  equals(abs(p.z - round(p.z)), 0.);
}
bool is_lattice(double x, double y, double z) {  // 未
    return equals(abs(x - round(x)), 0.) && equals(abs(y - round(y)), 0.) && equals(abs(z - round(z)), 0.);
}

bool is_parallel(Vector3d v1, Vector3d v2) {
    if (equals(v1.x*v2.y, v1.y*v2.x) && 
        equals(v1.y*v2.z, v1.z*v2.y) &&
        equals(v1.z*v2.x, v1.x*v2.z)) return true;
    return false;
}
bool is_orthogonal(Vector3d v1, Vector3d v2) {
    if (equals(v1.x*v2.x + v1.y*v2.y + v1.z*v2.z, 0.)) return true;
    return false;
}

struct Line3d {
    Point3d o, d; // p = o + k * d (k is a real parameter)

    Line3d() {}
    Line3d(Point3d p1, Point3d p2) : d(p2 - p1), o(p1) {}
    Line3d(double a, double b, double c, double e) { // ax + by + cz + e = 0.  未
        if (equals(a, 0)) o = Point3d{0, -e / b, 0}, d = Point3d{1, -a / b, 0};
        else if (equals(b, 0)) o = Point3d{-e / a, 0, 0}, d = Point3d{-b / a, 1, 0};
        else o = Point3d{0, 0, -e / c}, d = Point3d{1, 0, -a / c};
    }

    double dist2(Point3d p) { return dot(cross(d, p - o), cross(d, p - o)) / dot(d, d); }
    double dist(Point3d p) { return sqrt(dist2(p)); }
    bool cmp_proj(Point3d p, Point3d q) { return dot(d, p) < dot(d, q); }   

    friend istream& operator>>(istream &is, Line3d &L) {
        Point3d p1, p2;
        is >> p1 >> p2;
        L = Line3d(p1, p2);
		return is;
	}

    // bool operator==(const Line3d &L) const {}
};

struct Segment3d {
    Point3d p1, p2;

    Segment3d() {}
    Segment3d(Point3d p1, Point3d p2) : p1(p1), p2(p2) {}

    double length() { return dist(p1, p2); } 

    friend istream& operator>>(istream &is, Segment3d &S) {
		is >> S.p1 >> S.p2;
		return is;
	}

    bool operator==(const Segment3d &S) const {
        return (p1 == S.p1 && p2 == S.p2) or (p1 == S.p2 && p2 == S.p1);
    }
};

bool on_line(Line3d L, Point3d p) {
    return equals(abs(cross(L.o - p, L.o + L.d - p)), 0);
    // return is_parallel(L.o - p, L.o + L.d - p);
}
bool on_segment(Segment3d S, Point3d p) {
    if (!on_line(Line3d{S.p1, S.p2}, p)) return false;
    return equals(abs(S.p2 - S.p1), abs(p - S.p1) + abs(p - S.p2));
}

// small angle between direction vectors of two lines
double angle(Line3d L1, Line3d L2) {
    return small_angle(L1.d, L2.d);
}

Point3d projection(Line3d L, Point3d p) {
    Vector3d base = L.d;
    double t = dot(p - L.o, base) / norm(base);
    return L.o + base * t;
}
// Point3d projection(Segment3d S, Point3d p) { // 未. closet_point
// 	Vector base = S.p2 - S.p1;
// 	double r = dot(p - S.p1, base) / base.norm();
// 	Point3d proj = S.p1 + base * r;
//     if (r < 0.) return S.p1;
//     if (r > 1.) return S.p2;
//     return proj;
// }

Point3d reflection(Line3d L, Point3d p) {
    return p + (projection(L, p) - p) * 2.;
}

bool is_parallel(Line3d L1, Line3d L2) { return is_parallel(L1.d, L2.d); }
bool is_parallel(Line3d L, Segment3d S) { return is_parallel(L.d, S.p2 - S.p1); }
bool is_parallel(Segment3d S1, Segment3d S2) { return is_parallel(S1.p2 - S1.p1, S2.p2 - S2.p1); }

bool is_orthogonal(Line3d L1, Line3d L2) { return is_orthogonal(L1.d, L2.d); }
bool is_orthogonal(Line3d L, Segment3d S) { return is_orthogonal(L.d, S.p2 - S.p1); }
bool is_orthogonal(Segment3d S1, Segment3d S2) { return is_orthogonal(S1.p2 - S1.p1, S2.p2 - S2.p1); }

// bool intersect_LL(Line3d L1, Line3d L2) {
// 	return !is_parallel(L1, L2) && (同一平面上);
// }
// bool intersect_LS(Line3d L, Segment3d S) { // 2次元のもの
//     return (cross(L.d, S.p1 - L.p1) * cross(L.d, S.d) < EPS);
// }
// bool intersect_SS(Segment3d S1, Segment3d S2) {
// }

// vector<Point3d> cross_point_LL(Line3d L1, Line3d L2) {}
// vector<Point3d> cross_point_LS(Line3d L, Segment3d S) {}
// vector<Point3d> cross_point_SS(Segment3d S1, Segment3d S2) {}

double dist_Lp(Line3d L, Point3d p) {
    return abs(cross(L.d, p - L.o)) / abs(L.d);
}

double dist_Sp(Segment3d S, Point3d p) {
    Point3d r = projection(Line3d{S.p1, S.p2}, p);
    if (on_segment(S, r)) return abs(p - r);
    return min(abs(S.p1 - p), abs(S.p2 - p));
}

pair<Point3d, Point3d> closest_pair(Line3d L1, Line3d L2) {
    Point3d d1 = normalize(L1.d);
    Point3d d2 = normalize(L2.d);
    // P(x) = L1.o + d1 * x, Q(y) = L2.o + d2 * y. min |P(x) - Q(y)| is the distance of the lines L1, L2.
    double p = dot(L2.d, d1), q = dot(L2.d, d2), c = dot(d1, d2);
    // x - cy = p
    // cx - y = q
    double x, y;
    if (equals(abs(c), 1.)) { // parallel <=> d1 // d2
        x = p, y = 0;
    } else {
        double den = 1 - c*c;
        x = (p - c*q) / den, y = (c*p - q) / den;
    }
    return {L1.o + d1*x, L2.o + d2*y};
} // ライブラリにほかの実装あり
// pair<Point3d, Point3d> closest_pair(Segment3d L1, Segment3d L2) {}

double dist_LL(Line3d L1, Line3d L2) {
    auto [p, q] = closest_pair(L1, L2);
    return dist(p, q);
}
double dist_LS(Line3d L, Segment3d S) {
    auto [p, q] = closest_pair(L, Line3d{S.p1, S.p2});
    return on_segment(S, q) ? dist(p, q) : min(dist_Lp(L, S.p1), dist_Lp(L, S.p2));
}
double dist_SS(Segment3d S1, Segment3d S2) {
    auto [p, q] = closest_pair(Line3d{S1.p1, S1.p2}, Line3d{S2.p1, S2.p2});
    return on_segment(S1, p) && on_segment(S2, q) ? 
        dist(p, q) : 
        min({dist_Sp(S1, S2.p1), dist_Sp(S1, S2.p2), dist_Sp(S2, S1.p1), dist_Sp(S2, S1.p2)});
}

struct Plane {
    Point3d n; // 法線ベクトル
    double d;  // dot(n, p)

    Plane() {}
    Plane(Point3d n, double d) : n(n), d(d) {}
    Plane(Point3d n, Point3d p) : n(n), d(dot(n, p)) {}
    Plane(Point3d p, Point3d q, Point3d r) : Plane(cross(q - p, r - p), p) {} // non-collinear points P,Q,R

    // 正:nと同じ側, 0:面上, 負:nと反対側
    double side(Point3d p) { return dot(n, p) - d; }
    
    // tだけ平行移動
    Plane translate(Point3d t) { return {n, d + dot(n, t)}; }
    // nの方向にdだけ平行移動
    Plane shiftUp(double dist) { return {n, d + dist * abs(n)}; }

    // 平面上の異なる点を2つ返す
    pair<Point3d, Point3d> get_two_points_on_plane() {
        assert(sign(n.x) != 0 or sign(n.y) != 0 or sign(n.z) != 0);
        if (sign(n.x) == 0 && sign(n.y) == 0) return {Point3d(1, 0, d/n.z),  Point3d(0, 1, d/n.z)};
        if (sign(n.y) == 0 && sign(n.z) == 0) return {Point3d(d/n.x, 1, 0),  Point3d(d/n.x, 0, 1)};
        if (sign(n.z) == 0 && sign(n.x) == 0) return {Point3d(1, d/n.y, 0),  Point3d(0, d/n.y, 1)};
        if (sign(n.x) == 0) return {Point3d(1, d/n.y, 0),  Point3d(0, 0, d/n.z)};
        if (sign(n.y) == 0) return {Point3d(0, 1, d/n.z),  Point3d(d/n.x, 0, 0)};
        if (sign(n.z) == 0) return {Point3d(d/n.x, 0, 1),  Point3d(0, d/n.y, 0)};
        if (sign(d)!=0) return {Point3d(d/n.x, 0, 0),  Point3d(0, d/n.y, 0)};
        return {Point3d(n.y, -n.x, 0),  Point3d(-n.y, n.x, 0)};
    }
};

Point3d projection(Plane P, Point3d p) {
    Point3d a = P.n * P.d;
    return p - (P.n * dot(p - a, P.n));
}
Point3d reflection(Plane P, Point3d p) { 
    return p + (projection(P, p) - p) * 2.;
}

bool is_parallel(Plane P, Line3d L) { return is_orthogonal(P.n, L.d); }
bool is_parallel(Plane P, Segment3d S) { return is_orthogonal(P.n, S.p1 - S.p2); }
bool is_orthogonal(Plane P, Line3d L) { return is_parallel(P.n, L.d); }
bool is_orthogonal(Plane P, Segment3d S) { return is_parallel(P.n, S.p1 - S.p2); }

bool is_parallel(Plane P1, Plane P2) { // (0, 0, 0)か
    return abs(cross(P1.n, P2.n)) < EPS;
}
bool is_orthogonal(Plane P1, Plane P2) { return sign(dot(P1.n, P2.n)) == 0; }

double angle(Plane P, Line3d l) {
    return PI / 2. - acos(min(fabs(dot(P.n, l.d)) / abs(P.n) / abs(l.d), 1.));
}

bool intersect_PL(Plane P, Line3d L) { return !is_parallel(P, L); }
bool intersect_PS(Plane P, Segment3d S) {
    Point3d a = P.n * P.d;
    double b = dot(a - S.p1, P.n);
    double c = dot(a - S.p2, P.n);
    if (b > c) swap(b, c);
    if (b < EPS && c > -EPS) return true;
    return false;
}
bool intersect_PP(Plane P1, Plane P2) { return !is_parallel(P1, P2); }

Point3d cross_point_PL(Plane P, Line3d L) { 
    return L.o - L.d * P.side(L.o) / dot(L.d, P.n); 
}
Point3d cross_point_PS(Plane P, Segment3d S) {
    Point3d a = P.n * P.d;
    double dot_p0a = fabs(dot(S.p1 - a, P.n));
    double dot_p1a = fabs(dot(S.p2 - a, P.n));
    if (equals(dot_p0a + dot_p1a, 0)) return S.p1;
    return S.p1 + (S.p2 - S.p1) * (dot_p0a / (dot_p0a + dot_p1a));
}
Line3d cross_line_PP(Plane p1, Plane p2) { // p1とp2は同一でない
    Point3d d = cross(p1.n, p2.n);
    Point3d o = cross(p2.n * p1.d - p1.n * p2.d, d) / dot(d, d);
    return {o, d};
}

double dist_Pp(Plane P, Point3d p) { return fabs(P.side(p)) / abs(P.n); }
// https://yukicoder.me/problems/no/132
// double dist_Pp(Plane P, Point3d p) {
//     Point3d a = P.n * P.d;  // 平面上の適当な点をつくる
//     return abs(dot(p - a, P.n));
// } // https://yukicoder.me/problems/no/132
double dist_PL(Plane P, Line3d L) { return is_parallel(P, L) ? dist_Pp(P, L.o) : 0; }
double dist_PS(Plane P, Segment3d S) {
    Point3d ha = projection(P, S.p1), hb = projection(P, S.p2);
    double ipa = dot(S.p1 - ha, S.p2 - S.p1), ipb = dot(S.p2 - hb, S.p1 - S.p2);
    return sign(ipa) < 0 && sign(ipb) < 0 ? 0 : min(dist_Pp(P, S.p1), dist_Pp(P, S.p2));
}
double dist_PP(Plane P1, Plane P2) { // 未
    if (!is_parallel(P1, P2)) return 0;
    return abs(P1.d - P2.d) / abs(P1.n);
}

using P3db = pair<Point3d, bool>;

double area(Point3d a, Point3d b, Point3d c) {
    return abs(cross(b - a, c - a)) / 2.;
}

signed main() {
    int N;
    cin >> N;
    Point3d p;
    cin >> p;
    
    vector<Point3d> P(N);
    for (int i = 0; i < N; i++) {
        cin >> P[i];
    }
    
    double ans = 0;
    for (int i = 0; i < N; i++) {
        for (int j = i + 1; j < N; j++) {
            for (int k = j + 1; k < N; k++) {
                Plane f(P[i], P[j], P[k]);
                ans += dist_Pp(f, p);
            }
        }
    }
    cout << ans << endl;
}