#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

template <class S, class Coordinate> class rangetree {
    int n;
    using Pt = pair<Coordinate, Coordinate>;
    vector<Pt> _pts;
    vector<vector<Pt>> _range2yxs;
    vector<fenwick_tree<S>> bit; // 乗せたいBITを書く
    void _set(int v, Pt p, S val) {
        auto i = distance(
            _range2yxs[v].begin(),
            lower_bound(_range2yxs[v].begin(), _range2yxs[v].end(), Pt{p.second, p.first}));
        bit[v].add(i, -bit[v].sum(i, i + 1) + val);
    }
    void _add(int v, Pt p, S val) {
        auto i = distance(
            _range2yxs[v].begin(),
            lower_bound(_range2yxs[v].begin(), _range2yxs[v].end(), Pt{p.second, p.first}));
        bit[v].add(i, val);
    }
    S _prod(int v, Coordinate yl, Coordinate yr) {
        auto comp = [&](const Pt &l, const Pt &r) { return l.first < r.first; };
        int il = distance(
            _range2yxs[v].begin(),
            lower_bound(_range2yxs[v].begin(), _range2yxs[v].end(), Pt{yl, yl}, comp));
        int ir = distance(
            _range2yxs[v].begin(),
            lower_bound(_range2yxs[v].begin(), _range2yxs[v].end(), Pt{yr, yr}, comp));
        return bit[v].sum(il, ir);
    }

public:
    rangetree() = default;
    void add_point(Coordinate x, Coordinate y) noexcept { _pts.emplace_back(x, y); }
    void build() {
        sort(_pts.begin(), _pts.end());
        _pts.erase(unique(_pts.begin(), _pts.end()), _pts.end());
        n = _pts.size();

        _range2yxs.resize(n * 2);
        for (int i = 0; i < n; i++) _range2yxs[n + i] = {{_pts[i].second, _pts[i].first}};
        for (int i = n - 1; i > 0; i--) {
            auto &lch = _range2yxs[i * 2];
            auto &rch = _range2yxs[i * 2 + 1];
            merge(
                lch.begin(), lch.end(), rch.begin(), rch.end(), back_inserter(_range2yxs[i]));
            _range2yxs[i].erase(
                unique(_range2yxs[i].begin(), _range2yxs[i].end()), _range2yxs[i].end());
        }
        for (const auto &v : _range2yxs) bit.emplace_back(v.size());
    }
    void set(Coordinate x, Coordinate y, S val) {
        int i = distance(_pts.begin(), lower_bound(_pts.begin(), _pts.end(), Pt{x, y}));
        assert(i < n && _pts[i] == make_pair(x, y));
        for (i += n; i; i >>= 1) _set(i, {x, y}, val);
    }
    void add(Coordinate x, Coordinate y, S val) {
        int i = distance(_pts.begin(), lower_bound(_pts.begin(), _pts.end(), Pt{x, y}));
        assert(i < n && _pts[i] == make_pair(x, y));
        for (i += n; i; i >>= 1) _add(i, {x, y}, val);
    }
    S prod(Coordinate xl, Coordinate xr, Coordinate yl, Coordinate yr) {
        auto comp = [](const Pt &l, const Pt &r) { return l.first < r.first; };
        int l = n + distance(_pts.begin(),
                                  lower_bound(_pts.begin(), _pts.end(), Pt{xl, yr}, comp));
        int r = n + distance(_pts.begin(),
                                  lower_bound(_pts.begin(), _pts.end(), Pt{xr, yr}, comp));
        S ret = 0;
        while (l < r) {
            if (l & 1) ret += _prod(l++, yl, yr);
            if (r & 1) ret += _prod(--r, yl, yr);
            l >>= 1, r >>= 1;
        }
        return ret;
    }
    S get(Coordinate x, Coordinate y) const { return prod(x, x + 1, y, y + 1); }
};

using S = modint1000000007;
rangetree<S, int> RT;
vector<int> A, dp, L, X;
vector<S> cnt;

signed main() {
    int N;
    cin >> N;
	A.resize(N);
	for (int i = 0; i < N; i++) cin >> A[i];

    Compress<int> C(A);
    A = C.zip(A);
	
    int len = 0;
    dp.assign(N + 1, N + 1);
    L.resize(N + 1);
    X.resize(N);
    cnt.resize(N + 1);

	for (int i = 0; i < N; i++) {
		int x = lower_bound(L.begin(), L.begin() + len, A[i]) - L.begin();
        X[i] = x;
        dp[i] = x + 1;
        L[x] = A[i]; // 長さxの最後尾の要素の最小値はA[i]
        if (dp[i] > len) len++;

        RT.add_point(A[i], dp[i]);
	}

    RT.build();

    dp.assign(N + 1, N + 1);
    L.assign(N + 1, 0);
    len = 0;

    for (int i = 0; i < N; i++) {
		int x = X[i];
        dp[i] = x + 1;
        L[x] = A[i]; // 長さxの最後尾の要素の最小値はA[i]
        if (dp[i] > len) len++;

        if (dp[i] > 0) {
            // 半開 x 半開
            // x1, x2, y1, y2
            cnt[i] += RT.prod(0, A[i], dp[i] - 1, dp[i]);
        }
        if (cnt[i].val() == 0) cnt[i] = 1;
        RT.add(A[i], dp[i], cnt[i]);
	}

    S ans = 0;
    for (int i = 0; i < N; i++) {
        if (dp[i] == len) ans += cnt[i];
    }
    cout << ans.val() << endl;
}