#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 Point {
	double x, y;

	Point() {}
	Point(double x, double y) : x(x), y(y) {}

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

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

	bool operator==(const Point &p) const { return (fabs(x - p.x) < EPS && fabs(y - p.y) < EPS); }
	// bool operator!=(const Point &p) const { return (fabs(x - p.x) > EPS or fabs(y - p.y) > EPS); }
	bool operator<(const Point &p) const { return (x != p.x ? x < p.x : y < p.y); } // (x, y) の辞書順比較

	double norm() { return x*x + y*y; }
	double abs() { return sqrt(norm()); }
};
typedef Point Vector;

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

double norm(Vector a) { return a.x*a.x + a.y*a.y; }
double abs(Vector a) { return sqrt(norm(a)); }
double dist(Point a, Point b) { return sqrt(norm(a - b)); }
double dot(Vector a, Vector b) { return a.x*b.x + a.y*b.y; }
double cross(Vector a, Vector b) { return a.x*b.y - a.y*b.x; }
Vector normalize(Vector a) { return a / abs(a); } // 長さを1に正規化

double rad_to_deg(double rad) { return rad * 180. / PI; }
double deg_to_rad(double deg) { return deg * PI / 180.; }

double angle(Vector a, Vector b) { return acos(dot(a, b) / (abs(a) * abs(b))); } // ベクトル間の角度(rad)
double arg(Vector p) { return atan2(p.y, p.x); } // 偏角(rad)
Point polar(double r, double rad) { return {cos(rad)*r, sin(rad)*r}; } // 極座標 to 直交座標

void arg_sort(vector<Point> &P) {
    auto sign_ = [&](const Point &p) -> int {
        if (abs(p.x) <= EPS && abs(p.y) <= EPS) return 0;
        else if (p.y < -EPS or abs(p.y) <= EPS && p.x > EPS) return -1;
        else return 1;
    };
    auto comp = [&](const Point &p, const Point &q) -> bool {
        return (sign_(p) != sign_(q) ? sign_(p) < sign_(q) : p.x * q.y - p.y * q.x > 0);
    };
    sort(P.begin(), P.end(), comp);
}

Vector orth(Vector p) { return {-p.y, p.x}; }
Point rot90(Point p) { return {-p.y, p.x}; }
Point rot180(Point p) { return {-p.x, -p.y}; }
Point rot(Point p, double theta) { return {cos(theta)*p.x - sin(theta)*p.y, sin(theta)*p.x + cos(theta)*p.y}; }
Point rot(Point p, Point c, double theta) { // 点pを、点cを中心として反時計回りにtheta(rad)回転(p != c の必要)
    double q = arg(p - c) + theta;
    return c + Point{cos(q)*abs(p - c), sin(q)*abs(p - c)};
}

bool is_lattice(Point p) { return equals(abs(p.x - round(p.x)), 0.) && equals(abs(p.y - round(p.y)), 0.); } // 未
bool is_lattice(double x, double y) { return equals(abs(x - round(x)), 0.) && equals(abs(y - round(y)), 0.); }

bool is_parallel(Vector a, Vector b) { return equals(cross(a, b), 0.); }
bool is_orthogonal(Vector a, Vector b) { return equals(dot(a, b), 0.); }

int ccw(Point p1, Point p2, Point p3) {
	Vector a = p2 - p1, b = p3 - p1;
	if (cross(a, b) > EPS) return 1; // 反時計
	if (cross(a, b) < -EPS) return -1; // 時計
	if (dot(a, b) < -EPS) return 2; // 真逆
	if (a.norm() < b.norm()) return -2; // Bの上にA
	return 0; // Aの上にB
}

struct Circle {
	Point c;
	double r;

	Circle() {}
	Circle(Point c, double r) : c(c), r(r) {}

	double area() { return PI * r * r; }

	bool operator == (const Circle &C) { return c == C.c && equals(r, C.r); }
    friend istream& operator>>(istream &is, Circle &C) {
		is >> C.c >> C.r;
		return is;
	}
};

typedef vector<Point> Polygon;

// TODO : angle(L1, L2), rot(L, p, theta), rot(S, p, theta), reflection(L, S), reflection(L, C), reflection(L, P)

struct Line {
	Point p1, p2;

	Line() {}
	Line(Point p1, Point p2) : p1(p1), p2(p2) {}
	// Ax + By = C
    Line(double A, double B, double C) {
        if (equals(A, 0)) p1 = Point(0, C / B), p2 = Point(1, C / B);
        else if (equals(B, 0)) p1 = Point(C / A, 0), p2 = Point(C / A, 1);
        else p1 = Point(0, C / B), p2 = Point(C / A, 0);
    }

	// tuple<int, int, int> get_abc() {}

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

	bool operator==(const Line& l) const { 
        if (abs(cross(p1 - l.p1, l.p2 - l.p1)) > EPS) return false;
        if (abs(cross(p2 - l.p1, l.p2 - l.p1)) > EPS) return false;
        return true;
    }
	// bool operator<(const Line& l) const { return (l.p2.y - l.p1.y) * (p2.x - p1.x) < (l.p2.x - l.p1.x) * (p2.y - p1.y); }
};

struct Segment {
	Point p1, p2;

	Segment() {}
	Segment(Point p1, Point p2) : p1(p1), p2(p2) {}

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

	// 未
    bool operator==(const Segment& s) const { return (abs(p1 - s.p1) < EPS && abs(p2 - s.p2) < EPS); }
	bool operator!=(const Segment& s) const { return (abs(p1 - s.p1) > EPS or abs(p2 - s.p2) > EPS); }
};

bool on_line(Line L, Point p) {
	if (sign(cross(L.p1 - p, L.p2 - p)) != 0) return false;
	return true;
}
bool on_segment(Segment S, Point p) {
	if (sign(cross(S.p1 - p, S.p2 - p)) != 0) return false;
  	if (sign(dot(S.p1 - p, S.p2 - p)) > 0) return false;
	return true;
}

int Location(Line L, Point p) { return ccw(L.p1, p, L.p2); } // -1, 1, else

Point projection(Line L, Point p) {
	Vector base = L.p2 - L.p1;
	double r = dot(p - L.p1, base) / base.norm();
	return L.p1 + base * r;
}
Point projection(Segment S, Point p) { // 未
	Vector base = S.p2 - S.p1;
	double r = dot(p - S.p1, base) / base.norm();
	Point proj = S.p1 + base * r;
    if (r < 0.) return S.p1;
    if (r > 1.) return S.p2;
    return proj;
}

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

bool is_parallel(Line L1, Line L2) { return is_parallel(L1.p2 - L1.p1, L2.p2 - L2.p1); }
bool is_parallel(Line L, Segment S) { return is_parallel(L.p2 - L.p1, S.p2 - S.p1); }
bool is_parallel(Segment S, Line L) { return is_parallel(S.p2 - S.p1, L.p2 - L.p1); }
bool is_parallel(Segment S1, Segment S2) { return is_parallel(S1.p2 - S1.p1, S2.p2 - S2.p1); }

bool is_orthogonal(Line L1, Line L2) { return is_orthogonal(L1.p2 - L1.p1, L2.p2 - L2.p1); }
bool is_orthogonal(Line L, Segment S) { return is_orthogonal(L.p2 - L.p1, S.p2 - S.p1); }
bool is_orthogonal(Segment S, Line L) { return is_orthogonal(S.p2 - S.p1, L.p2 - L.p1); }
bool is_orthogonal(Segment S1, Segment S2) { return is_orthogonal(S1.p2 - S1.p1, S2.p2 - S2.p1); }

bool is_same_line(Line L1, Line L2) { 
    if (abs(cross(L1.p1 - L2.p1, L2.p2 - L2.p1)) > EPS) return false;
    if (abs(cross(L1.p2 - L2.p1, L2.p2 - L2.p1)) > EPS) return false;
    return true;
}
// bool is_same_line(Line L1, Segment S2) { return is_same_line(L1.p2 - L1.p1, S2.p2 - S2.p1); }
// bool is_same_line(Segment S1, Line S2) { return is_same_line(S1.p2 - S1.p1, S2.p2 - S2.p1); }
// bool is_same_line(Segment S1, Segment S2) { return is_same_line(S1.p2 - S1.p1, S2.p2 - S2.p1); }

bool intersect_LL(Line L1, Line L2) {
	return !is_parallel(L1, L2);
}
// bool intersect_LS(Line L, Segment S) {
// 	return cross(L.p2 - L.p1, S.p1 - L.p1) * cross(L.p2 - L.p1, S.p2 - L.p1) < EPS;
// }
bool intersect_SS(Segment S1, Segment S2) {
	return (ccw(S1.p1, S1.p2, S2.p1) * ccw(S1.p1, S1.p2, S2.p2) <= 0 &&
			ccw(S2.p1, S2.p2, S1.p1) * ccw(S2.p1, S2.p2, S1.p2) <= 0);
}

vector<Point> cross_point_LL(Line L1, Line L2) {
	auto a = cross(L2.p2 - L2.p1, L1.p2 - L1.p1);
	auto b = cross(L1.p1 - L2.p1, L1.p2 - L1.p1);
	if ( sign(a)) return {L2.p1 + (L2.p2 - L2.p1)*(b/a)}; // 1点で交わる
	if (!sign(b)) return {L2.p1, L2.p2};                  // 同じ直線(交点は無限個)
	return {};                                            // 異なる直線で平行
}

vector<Point> cross_point_LS(Line L1, Segment S1) {
	auto a = cross(S1.p2 - S1.p1, L1.p2 - L1.p1);
	auto b = cross(L1.p1 - S1.p1, L1.p2 - L1.p1);
	if (a < 0) { a *= -1; b *= -1; }
	if (sign(b) < 0 or sign(a - b) < 0) return {};            // 交点なし
	if (sign(a) != 0) return {S1.p1 + (S1.p2 - S1.p1)*(b/a)}; // 交差する
	if (sign(b) == 0) return {S1.p1, S1.p2};                  // 線分が直線に含まれる(交点は無限個)
	return {};                                                // 異なる線で平行
}

vector<Point> cross_point_SS(Segment S1, Segment S2) {
	auto a = cross(S1.p2 - S1.p1, S2.p2 - S2.p1);
	auto b = cross(S2.p1 - S1.p1, S2.p2 - S2.p1);
	auto c = cross(S1.p2 - S1.p1, S1.p1 - S2.p1);
	if (a < 0) { a = -a; b = -b; c = -c; }
	if (sign(b) < 0 or sign(a-b) < 0 or
		sign(c) < 0 or sign(a-c) < 0) return {};              // 交差しない
	if (sign(a) != 0) return {S1.p1 + (S1.p2 - S1.p1)*(b/a)}; // 平行でなく交差する
	vector<Point> ret;                                        // 同一直線上にあり、交差する
	auto insert_if_possible = [&](Point p) {
		for (auto q : ret) if (sign(dot(p - q, p - q)) == 0) return;
		ret.emplace_back(p);
	};
	if (sign(dot(S1.p1-S2.p1, S1.p2-S2.p1)) <= 0) insert_if_possible(S2.p1);
	if (sign(dot(S1.p1-S2.p2, S1.p2-S2.p2)) <= 0) insert_if_possible(S2.p2);
	if (sign(dot(S2.p1-S1.p1, S2.p2-S1.p1)) <= 0) insert_if_possible(S1.p1);
	if (sign(dot(S2.p1-S1.p2, S2.p2-S1.p2)) <= 0) insert_if_possible(S1.p2);
	return ret;
}

double dist_Lp(Line L, Point p) {
	return abs(cross(L.p2 - L.p1, p - L.p1) / abs(L.p2 - L.p1));
}
double dist_Sp(Segment S, Point p) {
	if (dot(S.p2 - S.p1, p - S.p1) < 0.) return abs(p - S.p1);
	if (dot(S.p1 - S.p2, p - S.p2) < 0.) return abs(p - S.p2);
	return dist_Lp(Line{S.p1, S.p2}, p);
}

double dist_LS(Segment S, Line L) {
	if (cross_point_LS(L, S).size()) return 0;
	// if (intersect_LS(L, S).size()) return 0;
	return min(dist_Lp(L, S.p1), dist_Lp(L, S.p2));
}

double dist_SS(Segment S1, Segment S2) {
	if (intersect_SS(S1, S2)) return 0.;
	return min({dist_Sp(S1, S2.p1), dist_Sp(S1, S2.p2),
			    dist_Sp(S2, S1.p1), dist_Sp(S2, S1.p2)});
}

// TODO : reflect(Circle C, Line L), 点pからC上の点の最遠点, 

// TODO : reflect(Circle C, Line L), 点pからC上の点の最遠点, 

Circle three_point_circle(Point p, Point q, Point r) { // 3点を通る円
    Point u = orth(q - p), v = r - p;
    Point o = (p + q + u*dot(r - q, v) / dot(u, v)) / 2;
    return {o, norm(p - o)};
}

// (OUT, ON, IN)
int contains(Circle C, Point p) {
    if (equals(abs(C.c - p), C.r)) return 1; // ON
    if (abs(C.c - p) < C.r) return 2; // IN
    return 0; // OUT
}

Point closet_point(Circle C, Point p) { // pがCの内部の場合は?
    Point v = p - C.c;
    return C.c + v * C.r / norm(v);
    // return C.c + v * C.r / abs(v);
}
// closet_point(C, L), closet_point(C, S) はライブラリ内にあり(未verify)

double dist_Cp(Circle C, Point p) {
    return max(abs(C.c - p) - C.r, 0.);
}
double dist_Cp2(Circle C, Point p) { // 輪っかと点の距離
    return abs(abs(C.c - p) - C.r);
}


vector<Point> cross_point_CL(Circle C, Line L) {
	Point u = L.p2 - L.p1, v = L.p1 - C.c;
	auto a = dot(u, u), b = dot(u, v)/a, t = (dot(v, v) - C.r*C.r)/a;
	auto det = b*b - t;
	if (sign(det) <  0) return {};           // 交差しない
	if (sign(det) == 0) return {L.p1 - u*b}; // 1点で接する
	return {L.p1 - u*(b + sqrt(det)),        // 2点で交差
			L.p1 - u*(b - sqrt(det))};
}

//-----------------------------------------------------------------------------
// intersection of Segment and Circle
// number of points:
//   0 ==> no cross_point_CL
//   1 ==> touch                     外接? 内接する場合は?
//   2 ==> contained                 内部に完全に含まれる?
//   3 ==> penetrating single side   端点の一方を内部に含み、もう一方を踏まない
//   4 ==> penetrating both sides    2点で交わる
// sorted from S.p1 to S.p2; usually, one would use ans[0] and ans.back().  (気を付ける)
//-----------------------------------------------------------------------------
vector<Point> cross_point_CS(Circle C, Segment S) { // 候補のコードが 円/円と線 にあり
	Point u = S.p2 - S.p1, v = S.p1 - C.c;
	auto a = dot(u, u), b = dot(u, v)/a, t = (dot(v, v) - C.r*C.r)/a;
	auto det = b*b - t;
	if (sign(det) < 0) return {};

	auto t1 = -b - sqrt(det), t2 = -b + sqrt(det);
	vector<Point> ps;
	auto insert_if_possible = [&](Point p) {
		for (auto q : ps) if (sign(dot(p - q, p - q)) == 0) return;
		ps.emplace_back(p);
	};
	if (sign(C.r - norm(S.p1 - C.c))  >= 0) insert_if_possible(S.p1);
	if (sign(t1) >= 0 && sign(1 - t1) >= 0) insert_if_possible(S.p1 + u*t1);
	if (sign(t2) >= 0 && sign(1 - t2) >= 0) insert_if_possible(S.p1 + u*t2);
	if (sign(C.r - norm(S.p2 - C.c))  >= 0) insert_if_possible(S.p2);
	return ps;
}

double dist_CL(Circle C, Line L) { // (未)
    if (cross_point_CL(C, L).size()) return 0;
    return dist_Lp(L, C.c) - C.r;
}
double dist_CL2(Circle C, Line L) { // 輪っかと直線の距離(未)
    return abs(dist_Lp(L, C.c) - C.r);
}

double dist_CS(Circle C, Segment S) { // (未)
    if (cross_point_CS(C, S).size()) return 0;
    return dist_Sp(S, C.c) - C.r;
}
double dist_CS2(Circle C, Segment S) { // 輪っかと線分の距離(未)
    return abs(dist_Sp(S, C.c) - C.r);
}

// 0:非交差, 1:外接, 2:交差
int count_intersections_CL(Circle C, Segment S) { // 未
    double d = dist_Sp(S, C.c);
	if (equals(d, C.r)) return 1;
	if (d > C.r + EPS) return 0;
	return 2;
}

// 0:非交差(外側), 1:真に内包, 2:1点で内接, 3:2点で内接,   
// 4:外接(接線), 5:外接(接点)  (点で接するが延長すると接線となる場合も5)
// 6:1回交差, 7:2回交差
int relation_CS(Circle C, Segment S) { // 未
	int a = contains(C, S.p1);
	int b = contains(C, S.p2);
	if (a == 2 && b == 2) return 1;
	if (a == 1 && b == 1) return 3;
	if (a && b) return 2;
	if (a or b) return 5;
	if (!a && b or a && !b) return 6;

	double d = dist_Sp(S, C.c);
	if (equals(d, C.r)) return 4;
	if (d > C.r + EPS) return 0;
	return 7;
}

int count_intersections_CS(Circle C, Segment S) { // 交点の個数 (未)
    int x = relation_CS(C, S);
    if (x == 7 or x == 3) return 2;
    if (x == 0 or x == 1) return 0;
    return 1;
}

// (-3,    -2,   -1,   0,    1,    2,    3)
// (非交差, 外包, 外接, 交差, 内接, 内包, 一致)
int relation_CC(Circle C1, Circle C2) {
    if (C1 == C2) return 3; // 一致
	if (C1.r < C2.r) swap(C1, C2);
	double d = abs(C1.c - C2.c);
	double r = C1.r + C2.r;
	if (equals(d, r)) return -1; // 外接
	if (d > r) return -3; // 非交差
	if (equals(d + C2.r, C1.r)) return 1; // 内接
	if (d + C2.r < C1.r) return 2; // 内包(外包TODO)
	return 0; // 交差
}

int count_intersections_CC(Circle C1, Circle C2) { // 交点の個数
    int x = relation_CC(C1, C2);
    if (x == 3) return INF;
    if (x == 0) return 2;
    if (x == -1 or x == 1) return 1;
    return 0;
} // https://www.acmicpc.net/problem/1002

int count_common_tangents_CC(Circle C1, Circle C2) { // 共通接戦の本数
	int x = relation_CC(C1, C2);
    if (x == 3) return INF;
    if (x == -3) return 4;
    if (x == -1) return 3;
    if (x == 0) return 2;
    if (x == 1) return 1;
	return 0;
} // https://onlinejudge.u-aizu.ac.jp/services/ice/?problemId=CGL_1_A

vector<Point> cross_point_CC(Circle C1, Circle C2) {
	if (C1.r < C2.r) swap(C1, C2);
	double g = dot(C1.c - C2.c, C1.c - C2.c);
	if (sign(g) == 0) {
		if (sign(C1.r - C2.r) != 0) return {};
		return {{C1.c.x + C1.r, C1.c.y}, {C1.c.x, C1.c.y + C1.r}, {C1.c.x - C1.r, C1.c.y}};
	}
	int inner = sign(g - (C1.r - C2.r)*(C1.r - C2.r));
	int outer = sign(g - (C1.r + C2.r)*(C1.r - C2.r));
	if (inner < 0 or outer > 0) return {};
	if (inner == 0) return {(C2.c*C1.r - C1.c*C2.r)/(C1.r - C2.r)};
	if (outer == 0) return {(C2.c*C1.r + C1.c*C2.r)/(C1.r + C2.r)};
	double eta = (C1.r*C1.r - C2.r*C2.r + g)/(g*2);
	double mu = sqrt(C1.r*C1.r/g - eta*eta);
	Point q = C1.c + (C2.c - C1.c)*eta, v = orth(C2.c - C1.c)*mu;
	return {q + v, q - v};
}

double dist_CC(Circle C1, Circle C2) {
    if (relation_CC(C1, C2) != -3) return 0;
    return abs(C1.c - C2.c) - C1.r - C2.r;
}

double dist_CC2(Circle C1, Circle C2) { // 輪っかと輪っかの距離
    int t = relation_CC(C1, C2);
    if (t == -1 or t == 0 or t == 1 or t == 3) return 0; // 内接, 交差, 外接, 一致
    if (t == -3) return abs(C1.c - C2.c) - C1.r - C2.r; // 非交差
    return abs(C1.r - C2.r) - abs(C1.c - C2.c); // 内包
}

vector<Point> tangentCp(Circle C, Point p) {
	return cross_point_CC(C, Circle(p, sqrt(norm(C.c - p) - C.r * C.r)));
}

vector<Line> tangentCC(Circle C1, Circle C2) {
	vector<Line> ls;
	if (C1.r < C2.r) swap(C1, C2);
	double g = abs(C1.c - C2.c);
	if (equals(g, 0.)) return ls;
	Point u = (C2.c - C1.c) / g;
	Point v = Point(-u.y, u.x);
	for (int s = 1; s >= -1; s -= 2) {
		double h = (C1.r + C2.r * s) / g;
		if (equals(1., h * h)) ls.push_back(Line(C1.c + u * C1.r, C1.c + (u + v) * C1.r));
		else if (1. - h * h > 0.) {
			Point uu = u * h, vv = v * sqrt(1. - h * h);
			ls.push_back(Line(C1.c + (uu + vv) * C1.r, C2.c - (uu + vv) * C2.r * s));
			ls.push_back(Line(C1.c + (uu - vv) * C1.r, C2.c - (uu - vv) * C2.r * s));
		}
	}
	return ls;
}


// 偏角ソート忘れに注意
// TODO : 一番上にある頂点のindex(二分探索)
//        intersect_PL_convex, cross_point_PS_convex, dist_PS_convex, cross_point_PP_convex
//        intersect_PL, cross_point_PL, cross_point_PS, tangent_Pp, dist_Pp, dist_PL, dist_PS, dist_PP, maximum_dist_PP, cross_point_PP
//        多角形の和, 共通部分, 差, 対称差

double perimeter(const Polygon &P) {
    double ret = 0;
    int N = P.size();
    for (int i = 0; i < N; i++) ret += dist(P[i], P[(i + 1) % N]);
    return ret;
}

double area(const Polygon &P) { // 符号付き
	double ret = 0;
	for (int i = 0; i < (int)P.size(); i++) {
		ret += cross(P[i], P[(i + 1) % P.size()]);
	}
	return ret / 2.;
}

Polygon to_rect(Point p1, Point p2) {
    return {p1, Point(p2.x, p1.y), p2, Point(p1.x, p2.y)};
}

void sort_ccw_convex(Polygon &P) {
    int N = P.size();
    double cx = 0, cy = 0; // 重心に対して偏角ソート
    for (const Point &p : P) {
        cx += p.x; cy += p.y;
    }
    cx /= N; cy /= N;

    sort(P.begin(), P.end(), [&](const Point &a, const Point &b) {
        auto sign_ = [&](const Point &p) -> int {
            if (abs(p.x - cx) <= EPS && abs(p.y - cy) <= EPS) return 0;
            else if (p.y - cy < -EPS or abs(p.y - cy) <= EPS && p.x - cx > EPS) return -1;
            else return 1;
        };
        return (sign_(a) != sign_(b) ? sign_(a) < sign_(b) : (a.x - cx) * (b.y - cy) - (a.y - cy) * (b.x - cx) > 0);
    });
} // 弱 https://yukicoder.me/problems/no/55

// boundaryは周上の点を含めるか否か. 基本的にはfalseにする。整数座標ok.
Polygon ConvexHull(Polygon P, bool boundary = false) {
	int N = P.size();
    if (N == 0) return {};

	sort(P.begin(), P.end(), [](const Point &a, const Point &b) {
			return (a.y != b.y ? a.y < b.y : a.x < b.x); });
    P.erase(unique(P.begin(), P.end()), P.end()); // 同じ座標の点を複数与えられる場合

    N = P.size();
    if (N == 1) return P; // 1点からなる場合(定義による)

	Polygon P2(2 * N);

	int k = 0;
	double e = boundary ? -EPS : +EPS;
	for (int i = 0; i < N; i++) {
		while (k > 1 && cross(P2[k - 1] - P2[k - 2], P[i] - P2[k - 1]) < e) k--;
		P2[k++] = P[i];
	}
	for (int i = N - 2, t = k; i >= 0; i--) {
		while (k > t && cross(P2[k - 1] - P2[k - 2], P[i] - P2[k - 1]) < e) k--;
		P2[k++] = P[i];
	}
	P2.resize(k - 1);
	return P2;
}

bool is_convex(const Polygon &P) {
	int N = P.size();
	for (int i = 0; i < N; i++) {
		if (ccw(P[(i - 1 + N) % N], P[i], P[(i + 1) % N]) == -1) return false;
    }
	return true;
}

int orientation(Point a, Point b, Point c) { return sign(cross(b - a, c - a)); }

// IN:2, ON:1, OUT:0
int contains(const Polygon &P, Point p) { // 凸性不要
	int N = P.size();
	bool x = false;
	for (int i = 0; i < N; i++) {
		Point a = P[i] - p;
		Point b = P[(i + 1) % N] - p;

		if (abs(cross(a, b)) < EPS && dot(a, b) < EPS) return 1;
		if (a.y > b.y) swap(a, b);
		if (a.y < EPS && EPS < b.y && cross(a, b) > EPS) x = !x;
	}
	return (x ? 2 : 0);
}
int contains_convex(const Polygon &P, Point p) { // O(logN)
    int N = P.size();
    int a = orientation(P[0], P[1], p), b = orientation(P[0], P[N - 1], p);
    if (a < 0 || b > 0) return 1;
    int l = 1, r = N - 1;
    while (l + 1 < r) {
        int mid = l + r >> 1;
        if (orientation(P[0], P[mid], p) >= 0) l = mid;
        else r = mid;
    }
    int k = orientation(P[l], P[r], p);
    if (k <= 0) return -k;
    if (l == 1 && a == 0) return 0;
    if (r == N - 1 && b == 0) return 0;
    return 2;
}

bool intersect_PS(const Polygon &P, Segment S) {
    int N = P.size();
    for (int i = 0; i < N; i++) {
        Segment S2(P[i], P[(i + 1) % N]);
        if (intersect_SS(S, S2)) return true;
    }
    return false;
}
bool intersect_PS_convex(const Polygon &P, Segment S) {
    int a = contains_convex(P, S.p1);
    int b = contains_convex(P, S.p2);
    return (a == 1 or b == 1 or a != b);
}

bool intersect_PP(const Polygon &P1, const Polygon &P2) { // O(NM)
    int N = P1.size();
    for (int i = 0; i < N; i++) {
        Segment S2(P1[i], P1[(i + 1) % N]);
        if (intersect_PS(P2, S2)) return true;
    }
    return false;
}
bool intersect_PP_convex(const Polygon &P1, const Polygon &P2) { // O(N logM)
    int N = P1.size();
    for (int i = 0; i < N; i++) {
        Segment S2(P1[i], P1[(i + 1) % N]);
        if (intersect_PS_convex(P2, S2)) return true;
    }
    return false;
}

// 凸多角形Pの頂点であって、点pとのdotが最大となるもののindex
// top - upper right vertex
// for minimum dot product negate p and return -dot(p, p[id])
int extreme_vertex(const Polygon &P, Point p, int top) { // O(log n)
    int N = P.size();
    if (N == 1) return 0;

	double ans = dot(P[0], p); 
    int id = 0;
    if (dot(P[top], p) > ans) ans = dot(P[top], p), id = top;

    int l = 1, r = top - 1;
    while (l < r) {
        int mid = l + r >> 1;
        if (dot(P[mid + 1], p) >= dot(P[mid], p)) l = mid + 1;
        else r = mid;
    }
    if (dot(P[l], p) > ans) ans = dot(P[l], p), id = l;

    l = top + 1, r = N - 1;
    while (l < r) {
        int mid = l + r >> 1;
        if (dot(P[(mid + 1) % N], p) >= dot(P[mid], p)) l = mid + 1;
        else r = mid;
    }
    l %= N;
    if (dot(P[l], p) > ans) ans = dot(P[l], p), id = l;
    return id;
}

// 凸多角形P, 直線L, Pの一番上の頂点(?)
// 交差する辺のindexを返す?
// it returns the indices of the edges of the polygon that are intersected by the line
// so if it returns i, then the line intersects the edge (P[i], P[(i + 1) % n])
vector<int> cross_point_PL_convex(const Polygon &P, Line L, int top) {
    Point a = L.p1, b = L.p2;
	int end_a = extreme_vertex(P, orth(a - b), top);
    int end_b = extreme_vertex(P, orth(b - a), top);
    auto cmp_l = [&](int i) { return orientation(a, P[i], b); };
    if (cmp_l(end_a) < 0 or cmp_l(end_b) > 0) return {}; // no intersection

    array<int, 2> ret;
    for (int i = 0; i < 2; i++) {
        int lo = end_b, hi = end_a, n = P.size();
        while ((lo + 1) % n != hi) {
            int m = ((lo + hi + (lo < hi ? 0 : n)) / 2) % n;
            (cmp_l(m) == cmp_l(end_b) ? lo : hi) = m;
        }
        ret[i] = (lo + !cmp_l(hi)) % n;
        swap(end_a, end_b);
    }
    if (ret[0] == ret[1]) return {ret[0]}; // touches the vertex ret[0]
    if (!cmp_l(ret[0]) && !cmp_l(ret[1])) 
        switch ((ret[0] - ret[1] + (int)P.size() + 1) % P.size()) {
            case 0: return {ret[0], ret[0]}; // touches the edge (ret[0], ret[0] + 1)
            case 2: return {ret[1], ret[1]}; // touches the edge (ret[1], ret[1] + 1)
        }
    return {ret[0], ret[1]}; // intersects the edges (ret[0], ret[0] + 1) and (ret[1], ret[1] + 1)
}

pair<Point, int> _tangent_Pp_convex(const Polygon &P, Point p, int dir, int l, int r) {
    while (r - l > 1) {
        int mid = (l + r) >> 1;
        bool pvs = orientation(p, P[mid], P[mid - 1]) != -dir;
        bool nxt = orientation(p, P[mid], P[mid + 1]) != -dir;
        if (pvs && nxt) return {P[mid], mid};
        if (!(pvs or nxt)) {
            auto p1 = _tangent_Pp_convex(P, p, dir, mid + 1, r);
            auto p2 = _tangent_Pp_convex(P, p, dir, l, mid - 1);
            return orientation(p, p1.first, p2.first) == dir ? p1 : p2;
        }
        if (!pvs) {
            if (orientation(p, P[mid], P[l]) == dir)  r = mid - 1;
            else if (orientation(p, P[l], P[r]) == dir) r = mid - 1;
            else l = mid + 1;
        }
        if (!nxt) {
            if (orientation(p, P[mid], P[l]) == dir)  l = mid + 1;
            else if (orientation(p, P[l], P[r]) == dir) r = mid - 1;
            else l = mid + 1;
        }
    }
    pair<Point, int> ret = {P[l], l};
    for (int i = l + 1; i <= r; i++) ret = orientation(p, ret.first, P[i]) != dir ? make_pair(P[i], i) : ret;
    return ret;
}

// ccw means the tangent from Q to that point is in the same direction as the polygon ccw direction
pair<int, int> tangent_Pp_convex(const Polygon &P, Point p) { // 接線となる点のindex
    int ccw = _tangent_Pp_convex(P, p, 1, 0, (int)P.size() - 1).second;
    int cw = _tangent_Pp_convex(P, p, -1, 0, (int)P.size() - 1).second;
    return make_pair(ccw, cw);
}

double dist_Pp_convex(const Polygon &P, Point p) {
    double ans = INF;
    int n = P.size();
    if (n <= 3) {
        for(int i = 0; i < n; i++) ans = min(ans, dist_Sp(Segment{P[i], P[(i + 1) % n]}, p));
        return ans;
    }
    auto [r, l] = tangent_Pp_convex(P, p);
    if (l > r) r += n;
    while (l < r) {
        int mid = (l + r) >> 1;
        double left = dot(P[mid % n] - p, P[mid % n] - p), right = dot(P[(mid + 1) % n] - p, P[(mid + 1) % n] - p);
        ans = min({ans, left, right});
        if (left < right) r = mid;
        else l = mid + 1;
    }
    ans = sqrt(ans);
    ans = min(ans, dist_Sp(Segment{P[l % n], P[(l + 1) % n]}, p));
    ans = min(ans, dist_Sp(Segment{P[l % n], P[(l - 1 + n) % n]}, p));
    return ans;
}

double dist_PL_convex(const Polygon &P, Line L, int top) { // O(log n)
	Point a = L.p1, b = L.p2;
	Point o = orth(b - a);
	if (orientation(a, b, P[0]) > 0) o = orth(a - b);
	int id = extreme_vertex(P, o, top);
	if (dot(P[id] - a, o) > 0) return 0.; //if o and a are in the same half of the line, then poly and line intersects
	return dist_Lp(L, P[id]); //does not intersect
}

double dist_PP_convex(const Polygon &P1, const Polygon &P2) { // O(N logN)
    double ans = INF;
    for (int i = 0; i < P1.size(); i++) {
        ans = min(ans, dist_Pp_convex(P2, P1[i]));
    }
    for (int i = 0; i < P2.size(); i++) {
        ans = min(ans, dist_Pp_convex(P1, P2[i]));
    }
    return ans;
}

double maximum_dist_PP_convex(Polygon P1, Polygon P2) { // O(N)
    int N = P1.size(), M = P2.size();
    double ans = 0;
    if (N < 3 or M < 3) {
        for (int i = 0; i < N; i++) {
            for (int j = 0; j < M; j++) ans = max(ans, dot(P1[i] - P2[j], P1[i] - P2[j]));
        }
        return sqrt(ans);
    }
    if (P1[0].x > P2[0].x) swap(N, M), swap(P1, P2);
    int i = 0, j = 0, step = N + M + 10;
    while (j + 1 < M && P2[j].x < P2[j + 1].x) j++ ;
    while (step--) {
        if (cross(P1[(i + 1) % N] - P1[i], P2[(j + 1) % M] - P2[j]) >= 0) j = (j + 1) % M;
        else i = (i + 1) % N;
        ans = max(ans, dot(P1[i] - P2[j], P1[i] - P2[j]));
    }
    return sqrt(ans);
}

// -3:非交差, -2:CがPに真に含まれる, -1:CがPに内接, 0:交差, 1:CがPに外接, 2:CがPを真に含む, 
// 3:辺で外接, 4:点で外接, 5:辺でも点でも外接
int relation_CP(Circle C, const Polygon &P) {
    int N = P.size();
    int in = 0, out = 0;
    bool r3 = false, r4 = false;
    for (int i = 0; i < N; i++) {
        int t = contains(C, P[i]);
        if (t == 2) in++;
        if (t == 0) out++;
        int r = relation_CS(C, Segment{P[i], P[(i + 1) % N]});
        if (r == 6 or r == 7) return 0; // 真に交差
        if (r == 4) r3 = true; // 辺で外接
        if (r == 5) r4 = true; // 点で外接
    }
    if (in == N) return 2;
    if (out == 0) return 1;

    if (contains(P, C.c) && out == N) return -2;
    if (contains(P, C.c) && in == 0) return -1;

    if (r3 && r4) return 5;
    if (r3) return 3;
    if (r4) return 4;
    return -3;
} 
// 弱(軸平行長方形, -2~2) https://atcoder.jp/contests/arc051/tasks/arc051_a
// 弱(三角形, -3~2) https://onlinejudge.u-aizu.ac.jp/services/ice/?problemId=0153

int count_intersections_CP(Circle C, const Polygon &P) { // 未
    int N = P.size();
    int ret = 0;
    // 円と線分集合と交点の個数 - 円上の点の個数
    for (int i = 0; i < N; i++) {
        ret += count_intersections_CS(C, Segment{P[i], P[(i + 1) % N]});
        ret -= (contains(C, P[i]) == 1);
    }
    return ret;
} // 弱(正方形) https://yukicoder.me/problems/no/1027

signed main() {
    double r;
    cin >> r;
    Circle C(Point{0., 0.}, sqrt(r));

    double d;
    cin >> d;
    d = sqrt(d);
    Polygon P = {Point{d, 0.}, Point{0., d}, Point{-d, 0.}, Point{0., -d}};

    cout << count_intersections_CP(C, P) << endl;
}