Problem:
Score: 500 points

Problem Statement

N people are standing in a queue, numbered 1, 2, 3, ..., N from front to back. Each person wears a hat, which is red, blue, or green.

The person numbered i says:

"In front of me, exactly A_i people are wearing hats with the same color as mine."

Assuming that all these statements are correct, find the number of possible combinations of colors of the N people's hats.

Since the count can be enormous, compute it modulo 1000000007.

Constraints

1 \leq N \leq 100000

0 \leq A_i \leq N-1

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
A_1 A_2 A_3 ... A_N

Output

Print the number of possible combinations of colors of the N people's hats, modulo 1000000007.

Sample Input 1

6
0 1 2 3 4 5

Sample Output 1

3

We have three possible combinations, as follows:

Red, Red, Red, Red, Red, Red

Blue, Blue, Blue, Blue, Blue, Blue

Green, Green, Green, Green, Green, Green

Sample Input 2

3
0 0 0

Sample Output 2

6

Sample Input 3

54
0 0 1 0 1 2 1 2 3 2 3 3 4 4 5 4 6 5 7 8 5 6 6 7 7 8 8 9 9 10 10 11 9 12 10 13 14 11 11 12 12 13 13 14 14 15 15 15 16 16 16 17 17 17

Sample Output 3

115295190

Solution in C++ (GCC 9.2.1) (judge verdict: Accepted)

#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
typedef long double ld;
const ll MOD = 1000000007;
// const ll MOD = 998244353;
const ll INF = MOD * MOD;
const long double EPS = 1e-12;

struct mint {
    ll x;
    mint(ll x = 0): x((x % MOD + MOD) % MOD){}
    mint operator-() const {return mint(-x);}

    mint& operator+=(const mint a){
        if ((x += a.x) >= MOD) x -= MOD;
        return *this;
    }

    mint& operator-=(const mint a){
        if ((x += MOD - a.x) >= MOD) x -= MOD;
        return *this;
    }

    mint& operator*=(const mint a) {(x *= a.x) %= MOD; return *this;}
    mint operator+(const mint a) const {return mint(*this) += a;}
    mint operator-(const mint a) const {return mint(*this) -= a;}
    mint operator*(const mint a) const {return mint(*this) *= a;}

    mint pow(ll t) const{
        if (!t) return 1;
        mint a = pow(t >> 1);
        a *= a;
        if (t & 1) a *= *this;
        return a;
    }

    // for prime mod
    mint inv() const {return pow(MOD - 2);}
    mint& operator/=(const mint a) {return *this *= a.inv();}
    mint operator/(const mint a) const {return mint(*this) /= a;}
};
istream& operator>>(istream& is, const mint& a) {return is >> a.x;}
ostream& operator<<(ostream& os, const mint& a) {return os << a.x;}

struct edge {
    ll to, cost;

    tuple<ll, ll> get_value(){return make_tuple(to, cost);}
};

struct fpoint {
    ld x = 0; ld y = 0;
    bool operator<(const fpoint& p) const{
        if (x == p.x) return y < p.y;
        return x < p.x;
    }

    bool operator==(const fpoint& p) const{
        if (p.x - EPS < x && x < p.x + EPS
            && p.y - EPS < y && y < p.y + EPS) return true;
        return false;
    }

    bool operator!=(const fpoint& p) const{
        if (p.x - EPS > x || x > p.x + EPS
            || p.y - EPS > y || y > p.y + EPS) return true;
        return false;
    }

    fpoint& operator+=(const ld a){x += a, y += a; return *this;}
    fpoint& operator-=(const ld a){x -= a, y -= a; return *this;}
    fpoint& operator*=(const ld a){x *= a, y *= a; return *this;}
    fpoint& operator/=(const ld a){x /= a, y /= a; return *this;}
    fpoint& operator+=(const fpoint p){x += p.x, y += p.y; return *this;}
    fpoint& operator-=(const fpoint p){x -= p.x, y -= p.y; return *this;}
    fpoint& operator*=(const fpoint p){x *= p.x, y *= p.y; return *this;}
    fpoint& operator/=(const fpoint p){x /= p.x, y /= p.y; return *this;}
    fpoint operator+(const fpoint p) const {return fpoint(*this) += p;}
    fpoint operator-(const fpoint p) const {return fpoint(*this) -= p;}
    fpoint operator*(const fpoint p) const {return fpoint(*this) *= p;}
    fpoint operator/(const fpoint p) const {return fpoint(*this) /= p;}
    fpoint operator+(const ld a) const {return fpoint(*this) += a;}
    fpoint operator-(const ld a) const {return fpoint(*this) -= a;}
    fpoint operator*(const ld a) const {return fpoint(*this) *= a;}
    fpoint operator/(const ld a) const {return fpoint(*this) /= a;}
    ld dot(const fpoint& p) const {return x * p.x + y * p.y;}
    ll cross(const fpoint& p) const {return x * p.y - y * p.x;}
    ld squared_norm() const {return x * x + y * y;}
    ld norm() const {return sqrt(x * x + y * y);}
    tuple<ld, ld> get_value(){return make_tuple(x, y);}

    ll which_quadrant() const{
        if (abs(x) < EPS && abs(y) < EPS) return 0;
        if (y > 0) return x > 0 ? 1 : 2;
        return x < 0 ? 3 : 4;
    }

    bool is_zero(){
        fpoint z = {0, 0};
        return z == *this;
    }

};

struct point {
    ll x = 0; ll y = 0;
    bool operator<(const point& p) const{
        if (x == p.x) return y < p.y;
        return x < p.x;
    }

    bool operator==(const point& p) const{
        if (x == p.x && y == p.y) return true;
        return false;
    }

    bool operator!=(const point& p) const{
        if (x != p.x || y != p.y) return true;
        return false;
    }

    point& operator+=(const ll a){x += a, y += a; return *this;}
    point& operator-=(const ll a){x -= a, y -= a; return *this;}
    point& operator*=(const ll a){x *= a, y *= a; return *this;}
    point& operator+=(const point p){x += p.x, y += p.y; return *this;}
    point& operator-=(const point p){x -= p.x, y -= p.y; return *this;}
    point& operator*=(const point p){x *= p.x, y *= p.y; return *this;}
    void operator++(int){x++, y++;}
    void operator++(){x++, y++;}
    void operator--(int){x--, y--;}
    void operator--(){x--, y--;}
    point operator+(const point p) const {return point(*this) += p;}
    point operator-(const point p) const {return point(*this) -= p;}
    point operator*(const point p) const {return point(*this) *= p;}
    point operator+(const ll a) const {return point(*this) += a;}
    point operator-(const ll a) const {return point(*this) -= a;}
    point operator*(const ll a) const {return point(*this) *= a;}
    ll dot(const point& p) const {return x * p.x + y * p.y;}
    ll cross(const point& p) const {return x * p.y - y * p.x;}
    ll squared_norm() const {return x * x + y * y;}
    tuple<ll, ll> get_value(){return make_tuple(x, y);}

    ll which_quadrant() const{
        if (x == 0 && y == 0) return 0;
        if (x >= 0 && y >= 0) return 1;
        else if (x <= 0 && y >= 0) return 2;
        else if (x <= 0 && y <= 0) return 3;
        else return 4;
    }

    bool is_zero(){
        point z = {0, 0};
        return z == *this;
    }

    fpoint to_fpoint(){
        fpoint ret = {ld(x), ld(y)};
        return ret;
    }
};

// angle_comparator
struct{
    template<typename T>
    bool operator()(const T p1, const T p2) const {
        ll q1 = p1.which_quadrant();
        ll q2 = p2.which_quadrant();
        if (q1 != q2) return q1 < q2;
        // judge for parallel lines
        // if p1 cross p2 > 0 -> sin arg(p1 -> o -> p2) > 0
        return p1.cross(p2) > 0;
    }
} angle_comparator;

struct undirected_edge {
    ll from; ll to; ll cost;
    bool operator<(const undirected_edge& ue) const{return cost < ue.cost;}
    tuple<ll, ll, ll> get_value(){return make_tuple(from, to, cost);}
};

struct event{
    ll loc, val, sgn;
    bool operator<(const event& e) const{
        if (loc == e.loc) return sgn == 1;
        return loc < e.loc;
    }
    
    bool operator>(const event& e) const{
        if (loc == e.loc) return sgn == -1;
        return loc > e.loc;
    }
    
    tuple<ll, ll, ll> get_value(){return make_tuple(loc, val, sgn);}
};

typedef std::pair<ll,ll> pl;
typedef std::tuple<ll, ll, ll> tp3;
typedef std::tuple<ll, ll, ll, ll> tp4;
typedef std::vector<ll> vl;
typedef std::vector<vl> vl2;
typedef std::vector<vl2> vl3;
typedef std::vector<vl3> vl4;
typedef std::vector<mint> vmi;
typedef std::vector<vmi> vmi2;
typedef std::vector<vmi2> vmi3;
typedef std::vector<vmi3> vmi4;
typedef std::vector<bool> vb;
typedef std::vector<vb> vb2;
typedef std::vector<vb2> vb3;
typedef std::vector<vb3> vb4;
typedef std::vector<pl> vpl;
typedef std::vector<tp3> vtp3;
typedef std::vector<tp4> vtp4;
typedef std::vector<point> points;
typedef std::vector<fpoint> fpoints;
// priority queue. Taking from the higher value. Don't forget calling !q.empty()
typedef std::priority_queue<ll> pq;
// priority queue. Taking from the lower value
typedef std::priority_queue<ll, vl, greater<ll>> pql;
typedef std::vector<vector<edge>> Graph;
const ll N_DIGITS = 60;
const long double PI = 3.14159265358979323846;
const points dirs = {{-1, 0}, { 1, 0}, { 0,  1}, {0, -1},  // four directions
                     { 1, 1}, {-1, 1}, {-1, -1}, {1, -1},  // diagonal
                     { 0, 0}};  // self

template <typename A, typename B>
string to_string(pair<A, B> p);
template <typename A, typename B, typename C>
string to_string(tuple<A, B, C> p);
template <typename A, typename B, typename C, typename D>
string to_string(tuple<A, B, C, D> p);
string to_string(const string& s) {return '"' + s + '"';}
string to_string(char c){return string(1, c);}
string to_string(const char* s) {return to_string((string) s);}
string to_string(bool b){return (b ? "true" : "false");}
string to_string(mint a){return to_string(a.x);}
string to_string(point p){return "{" + to_string(p.x) + ", " + to_string(p.y) + "}";}
string to_string(fpoint p){return "{" + to_string(p.x) + ", " + to_string(p.y) + "}";}
string to_string(edge e){return "{" + to_string(e.to) + ", " + to_string(e.cost) + "}";}
template <typename A, typename B>
string to_string(pair<A, B> p){return "(" + to_string(p.first) + ", " + to_string(p.second) + ")";}
template <typename A, typename B, typename C>
string to_string(tuple<A, B, C> p){return "(" + to_string(get<0>(p)) + ", " + to_string(get<1>(p)) + ", " + to_string(get<2>(p)) + ")";}
template <typename A, typename B, typename C, typename D>
string to_string(tuple<A, B, C, D> p){return "(" + to_string(get<0>(p)) + ", " + to_string(get<1>(p)) + ", " + to_string(get<2>(p)) + ", " + to_string(get<3>(p)) + ")";}
string to_string(vector<bool> v) {
  bool first = true;
  string res = "{";
  for (int i = 0; i < static_cast<int>(v.size()); i++) {
    if (!first) res += ", ";
    first = false;
    res += to_string(v[i]);
  }
  res += "}";
  return res;
}

template <size_t N>
string to_string(bitset<N> v) {
  string res = "";
  for (size_t i = 0; i < N; i++) res += static_cast<char>('0' + v[i]);
  return res;
}

template <typename A>
string to_string(A v) {
    bool first = true;
    string res = "{";
    for (const auto &x : v) {
        if (!first) {
            res += ", ";
        }
        first = false;
        res += to_string(x);
    }
    res += "}";
    return res;
}

template<typename T>
string to_string(priority_queue<T>& q){
    priority_queue<T> copy;
    bool first = true;
    string res = "{";
    while(!q.empty()){
        if (!first) {
            res += ", ";
        }
        first = false;
        res += to_string(q.top());
        copy.push(q.top());
        q.pop();
    }

    swap(q, copy);
    res += "}";
    return res;
}

template<typename T>
string to_string(queue<T>& q){
    queue<T> copy;
    bool first = true;
    string res = "{";
    while(!q.empty()){
        if (!first) {
            res += ", ";
        }
        first = false;
        res += to_string(q.front());
        copy.push(q.front());
        q.pop();
    }

    swap(q, copy);
    res += "}";
    return res;
}

void debug_out() {cerr << endl;}

template <typename Head, typename... Tail>
void debug_out(Head H, Tail... T){
    cerr << " " << to_string(H);
    debug_out(T...);
}

#define rep(i, n) for(ll (i) = 0; (i) < (n) ; (i)++ )
#define revrep(i, n) for(ll (i) = (n) - 1; (i) >= 0 ; (i)--)
#define For(i, a, b) for(ll (i) = (a); (i) < (b) ; (i)++ )
#define revFor(i, b, a) for(ll (i) = (b) - 1; (i) >= (a) ; (i)--)
#define all(a) (a).begin(), (a).end()
#define rall(a) (a).rbegin(), (a).rend()
#define isUpper(c) ('a' - (c) > 0)
#define isNum(c) (0 <= (c) - '0' && (c) - '0' <= 9)
#define toLower(c) char((c)+0x20)
#define toUpper(c) char((c)-0x20)
#define pb push_back
#define mp make_pair
#define mt make_tuple
#define pr(a) std::cout << (a)
#define prl(a) std::cout << (a) << endl
#define prl2(a,b) std::cout << (a) << " " << (b) << endl
#define prl3(a,b,c) std::cout << (a) << " " << (b) << " " << (c) << endl
#define prl4(a,b,c,d) std::cout << (a) << " " << (b) << " " << (c) << " " << (d) << endl
#define prs(a) std::cout << (a) << " "
#define prs2(a,b) std::cout << (a) << " " << (b) << " "
#define prs3(a,b,c) std::cout << (a) << " " << (b) << " " << (c) << " "
#define prs4(a,b,c,d) std::cout << (a) << " " << (b) << " " << (c) << " " << (d) << " "
#define yn(condition) if ((condition)) prl("Yes"); else prl("No");
#define YN(condition) if ((condition)) prl("YES"); else prl("NO");
#define in1(a) cin >> (a)
#define in2(a,b) cin >> (a) >> (b)
#define in3(a,b,c) cin >> (a) >> (b) >> (c)
#define in4(a,b,c,d) cin >> (a) >> (b) >> (c) >> (d)
#define in5(a,b,c,d,e) cin >> (a) >> (b) >> (c) >> (d) >> (e)
#define in6(a,b,c,d,e,f) cin >> (a) >> (b) >> (c) >> (d) >> (e) >> (f)
#define in7(a,b,c,d,e,f,g) cin >> (a) >> (b) >> (c) >> (d) >> (e) >> (f) >> (g)
#define e1 first
#define e2 second
#define popcount(i) __builtin_popcountl((i))
#define Forchar(c,a,z) for(char (c) = (a); (c) <= (z); (c)++)
#define cntchar(s,c) count(all((s)), c)
#define substring(s, start, end) s.substr((start), (end) - (start) + 1)
#define prl_nd(num, digits) std::cout << fixed << setprecision(digits) << (num) << endl;
#define prl_time(s) prl3("Elapsed Time:", 1000.0 * (clock() - s) / CLOCKS_PER_SEC, "[ms]");
#define char_to_str(c) string(1, (c))

#ifdef _LOCAL
#define debug(...) cerr << "[" << #__VA_ARGS__ << "]:", debug_out(__VA_ARGS__)
#else
#define debug(...) 42
#endif

ll modf(ll x, ll m){return (x % m + m) % m;}

template<class T>
inline bool chmax(T &a, T b) {
    if(a < b) {a = b; return true;}
    return false;
}

template<class T>
inline bool chmin(T &a, T b) {
    if(a > b){a = b; return true;}
    return false;
}

struct MaxFlow {
    struct F_edge {
        ll to, rev, capacity;
        F_edge(ll to, ll rev, ll capacity) : to(to), rev(rev), capacity(capacity) {}
    };

    typedef vector<F_edge> F_edges;
    vector<F_edges> graph;
    ll n_vertex;
    // level is the shortest path to get a given node from the source node.
    vl level, iter;

    MaxFlow(ll n_vertex) : n_vertex(n_vertex) {graph.resize(n_vertex);}

    void add_edge(ll from, ll to, ll capacity = 1) {
        graph[from].pb({to, ll(graph[to].size()), capacity});
        graph[to].pb({from, ll(graph[from].size()) - 1, 0});
    }

    void bfs(ll source){
        level = vl(n_vertex, -1); level[source] = 0;
        queue<ll> q; q.push(source);
        while (!q.empty()) {
            ll vertex = q.front(); q.pop();
            rep(i, graph[vertex].size()){
                ll target = graph[vertex][i].to;
                ll cap_target = graph[vertex][i].capacity;
                // if the flow can be into the target node, implement below.
                if (cap_target > 0 && level[target] < 0){
                    level[target] = level[vertex] + 1;
                    q.push(target);
                }
            }
        }
    }

    ll dfs(ll vertex, ll sink, ll flow) {
        if (vertex == sink) return flow;
        for(ll &i = iter[vertex]; i < graph[vertex].size(); i++){
            ll target = graph[vertex][i].to; ll cap_target = graph[vertex][i].capacity;
            ll rev_target = graph[vertex][i].rev;
            // if capasitiy is not full yet and target is farther,
            // then assign current flow.
            if (cap_target > 0 && level[vertex] < level[target]) {
                ll d = dfs(target, sink, min(cap_target, flow));
                if (d > 0){ // if the flow successfully reaches the sink, reduce the flow from the capacity 
                    graph[vertex][i].capacity -= d;
                    graph[target][rev_target].capacity += d;
                    return d;
                }
            }
        }
        return 0;
    }

    ll dinic(ll source, ll sink){
        // complexity O(EV^2)
        ll flow = 0;
        while (true) {
            bfs(source);
            // if there is no path leading to the sink, the maximum flow is 0.
            if (level[sink] < 0) return flow;
            iter = vl(n_vertex, 0); ll f;
            while ((f = dfs(source, sink, INF)) > 0) flow += f;
        }
    }
};

struct UnionFind {
    vl parents, set_size;
    set<ll> root_idx;
    ll n_groups;

    UnionFind(ll n){
        parents = set_size = vl(n);
        n_groups = n;
        rep(i, n){
            parents[i] = i;
            set_size[i] = 1LL;
            root_idx.insert(i);
        }
    }

    ll root_find(ll x){
        if (parents[x] == x) return x;
        return parents[x] = root_find(parents[x]);
    }

    void unite(ll x, ll y){
        // priority for x is larger than that of y
        x = root_find(x); y = root_find(y);
        if (x == y) return;
        parents[y] = x, set_size[x] += set_size[y];
        root_idx.erase(y);
        n_groups--;
    }

    // connected or not
    bool is_same(ll x, ll y){return root_find(x) == root_find(y);}
    ll size(ll x){return set_size[root_find(x)];}
};

struct WeightedUnionFind{
    vl parents, set_size, weights;
    set<ll> root_idx;
    ll n_groups;

    WeightedUnionFind(ll n){
        parents = set_size = weights = vl(n);
        n_groups = n;
        rep(i, n){
            parents[i] = i;
            set_size[i] = 1LL;
            weights[i] = 0;
            root_idx.insert(i);
        }
    }

    ll root_find(ll x){
        if (parents[x] == x) return x;
        ll r = root_find(parents[x]);
        weights[x] += weights[parents[x]];
        return parents[x] = r;
    }

    ll weight(ll x){
        root_find(x);
        return weights[x];
    }

    void unite(ll x, ll y, ll w){
        // priority for x is larger than that of y
        w += weight(x) - weight(y);
        x = root_find(x); y = root_find(y);
        if (x == y) return;
        parents[y] = x, set_size[x] += set_size[y];
        root_idx.erase(y);
        weights[y] = w;
        n_groups--;
    }

    // connected or not
    bool is_same(ll x, ll y){return root_find(x) == root_find(y);}
    ll size(ll x){return set_size[root_find(x)];}
    ll difference(ll x, ll y){return weight(y) - weight(x);}
};

struct Doubling { // ABC167D
    ll n;
    ll sz;
    vl2 next;
    /*
        next[k + 1][i] := next[k][next[k][i]]
        next[0][i] := edge[i]
        e.g. a0, a1, ..., an-1 / 0 <= ai <= n - 1
        a0 -> a[a0] -> a[a[a0]] -> ... -> a[a[...[a[0]]]] (m times)
        Let the function repeatedly inputting a[i] m times be f[m](a[i])

        - get(i, x) returns f[x](a[i])
        - lower_bound(i, j) returns minimum x which satisfies f[x](a[i]) >= j.
                                    if not possible returns n.

    */

    // edge[i]: i -> edge[i] by one operation
    Doubling(vl& edge) : n(edge.size()), sz(62) {
        next.resize(sz, vl(n, -1));
        rep(i, n) next[0][i] = edge[i];
        rep(k, sz - 1) rep(i, n) next[k + 1][i] = next[k][next[k][i]];
    }

    // return the result obtained when starting from i and advancing x times
    ll get(ll i, ll x){
        ll ret = i;
        rep(bit, sz){
            if (!(x >> bit & 1)) continue;
            ret = next[bit][ret];
        }
        return ret;
    }

    // return the iteration x where the value exceeds j for the first time.
    // if it is not possible, then return n.
    ll lower_bound(ll i, ll j) {
        ll cur = i, acc = 0;
        revrep(wid, sz){
            if (next[wid][cur] < j) {
                acc += 1LL << wid;
                cur = next[wid][cur];
            }
        }
        return min(n, acc + 1);
    }
};

template<class T>
class LowestCommonAncestor{
public:
    ll N, logN;
    vl depth, len;
    T tree;
    vl2 parents;

    LowestCommonAncestor(ll n, T& _tree){
        N = n;
        logN = 0;
        while (N > (1LL << logN)) logN++;
        depth = len = vl(N);
        parents = vl2(logN, vl(N));
        tree = _tree;
        _init(0, -1, 0, 0);
        _build();
    }

    void _init(ll source, ll parent, ll d, ll l){
        depth[source] = d;
        parents[0][source] = parent;
        len[source] = l;
        rep(i, tree[source].size()){
            ll target = tree[source][i].to;
            ll cost = tree[source][i].cost;
            if (target == parent) continue;
            _init(target, source, d + 1, cost + l);
        }
    }

    void _build(){
        rep(k, logN - 1) rep(n, N){
            // if there is no parent, -1.
            // otherwise, the parent of the parent is the parent.
            if (parents[k][n] < 0) parents[k + 1][n] = -1;
            else parents[k + 1][n] = parents[k][parents[k][n]];
        }
    }

    ll query(ll u, ll v){
        if (depth[u] > depth[v]) swap(u, v);
        rep(k, logN) if ((depth[v] - depth[u]) >> k & 1) v = parents[k][v];
        if (u == v) return u;

        revrep(k, logN){
            if (parents[k][u] != parents[k][v]){
                u = parents[k][u]; v = parents[k][v];
            }
        }
        return parents[0][u];
    }

    ll distance(ll u, ll v){
        ll w = query(u, v);
        return len[u] + len[v] - 2 * len[w];
    }
};

struct BinaryIndexedTree{
    ll n, ini, M; vl dat;
    BinaryIndexedTree(ll n, ll ini = 0):
        dat(n + 1, ini), n(n), ini(ini){M = 1; while (M <= n) M <<= 1;};

    //       x: 1001 1010  1100 1011 1101  1111
    // x & - x: 0001 0010  0100 0001 0001  0001
    //      ->: 1010 1100 10000 1100 1100 10000

    ll update_func(ll val, ll d){
        // if maximum -> max(val, dat)
        // return max(val, d);
        // if cumulative sum
        return val + d;
    }

    ll query(ll i){
        /*
        v[0] + v[1] + ... + v[i]
        e.g.) i = 10101
        itr1. 10101 -> 10100
        itr2. 10100 -> 10000
        itr3. 10000 -> 00000 (break)
        */

        if (i < 0) return ini;
        ll ret = 0;
        for (ll j = i; j >= 0; j = (j & (j + 1)) - 1){
            ret = update_func(ret, dat[j]);
        }
        return ret;
    }

    ll query(ll l, ll r){
        // a[l] + a[l + 1] + ... + a[r - 1] + a[r]
        return query(r) - query(l - 1);
    }

    ll lower_bound(ll key){
        //v[0] + v[1] + ... + v[left - 1] < key <= v[0] + v[1] + ... + v[left]
        if (key <= 0) return 0;
        ll left = 0, right = 1;
        while(right <= n) right <<= 2;

        for (ll i = right; i > 0; i /= 2){
            if (left + i <= n && dat[left + i - 1] < key){
                key -= dat[left + i - 1];
                left += i;
            }
        }
        return left;
    }

    void update(ll i, ll val){
        /*
        e.g.) i = 10101, n = 11111
        itr1. i: 10101, i+1: 10110 -> 10111
        itr2. i: 10111, i+1: 11000 -> 11111 (break)
        */
        if (i < 0) return;
        for(ll j = i; j < n; j |= j + 1){
            dat[j] = update_func(val, dat[j]);
        }
    }
};

struct SegmentTree{
	ll n, ini, minimize;
	vl dat;

    // when seeking minimum
    // ini = INF
    // when seeking maximum
    // ini = -INF
	SegmentTree(ll n_, bool minimize_ = true){
        n = 1; minimize = minimize_;
        if (minimize) ini = INF;
        else ini = - INF;
		while (n < n_) n *= 2;
		dat.resize(2 * n - 1);
		rep(i, 2 * n - 1) dat[i] = ini;
    };

	void update(ll idx, ll val, bool force = false){
		idx += n - 1;
        if (!force){
            if (minimize && dat[idx] <= val) return;
            if (!minimize && dat[idx] >= val) return;
        }

		dat[idx] = val;
		while (idx > 0){
			idx = (idx - 1) / 2;
            // when seeking minimum
			if (minimize) dat[idx] = min(dat[idx * 2 + 1], dat[idx * 2 + 2]);
            // when seeking maximum
            else dat[idx] = max(dat[idx * 2 + 1], dat[idx * 2 + 2]);
		}
	}

	ll query(ll l, ll r){
        // ### NOTE ###
        // the range is [l, r]
        // l, l + 1, ..., r
        r++; // to adjust to this method
        return query_segment(l, r, 0, 0, n);
	}

	ll query_segment(ll a, ll b, ll idx, ll l, ll r){
        assert(a <= b);

		if (r <= a || b <= l) return ini;
		if (a <= l && r <= b) return dat[idx];
		else{
            ll seg1 = query_segment(a, b, idx * 2 + 1, l, (l + r) / 2);
            ll seg2 = query_segment(a, b, idx * 2 + 2, (l + r) / 2, r);
            // when seeking minimum
            if (minimize) return min(seg1, seg2);
            // when seeking maximum
            else return max(seg1, seg2);
        }
	}
};

template<class Target>
class RerootingTreeDP {
public:
    using T = typename Target::type;
    struct DP_edge {
        ll to, rev; // rev is the index to trace the source node.
        T value; // objective value
    };
private:
    ll n;
    void dfs_fwd(ll source, ll parent){
        ll par_idx = -1;
        vector<T> values;
        rep(i, tree[source].size()) {
            const DP_edge& e = tree[source][i];
            if (e.to == parent){par_idx = i; continue;}
            dfs_fwd(e.to, source);
            values.pb(e.value);
        }

        // If the parent != -1, update the value on edge from parent to source
        if (par_idx != -1){
            ll src_idx = tree[source][par_idx].rev;
            // update values on the edge from parent to source
            tree[parent][src_idx].value = Target::merge(values);
        }
    }

    void dfs_bwd(ll source, ll parent){
        vector<T> values;
        for (auto&& e: tree[source]) values.pb(e.value);
        values = Target::evaluate(values);

        rep(i, tree[source].size()) {
            const DP_edge& e = tree[source][i];
            if (e.to == parent) continue;
            // tree[e.to][e.rev]: e.to -> source
            tree[e.to][e.rev].value = values[i];
            dfs_bwd(e.to, source);
        }
    }
public:
    UnionFind uf;
    vector<vector<DP_edge>> tree;
    RerootingTreeDP(ll n) : n(n), uf(n), tree(n) {}

    void add_edge(ll u, ll v, T val){
        assert(!uf.is_same(u, v));
        tree[u].pb({v, ll(tree[v].size()), val});
        tree[v].pb({u, ll(tree[u].size()) - 1, val});
        uf.unite(u, v);
    }

    void dp() {
        vb visited(n, false);
        rep(i, n){
            if (visited[uf.root_find(i)]) continue;
            dfs_fwd(i, -1);
            visited[uf.root_find(i)] = true;
        }

        visited.assign(n, false);
        rep(i, n){
            if (visited[uf.root_find(i)]) continue;
            dfs_bwd(i, -1);
            visited[uf.root_find(i)] = true;
        }
    }

    ll size() const {return tree.size();}
};

// ABC160F is one example
// Modify the functions evaluate and merge based on given problems
struct Merger{
    using type = ll;

    // This is the exaple of the number of children
    static type merge(const vector<type>& value){
        // merge the result below the source node
        // each value is from each child node of the source node
        // value[i] := f(child i)
        // Here, we would like to obtain f(source) using f(child i) (i = 0, 1, ..., n_children)
        ll ret = 1;
        for (auto&& v: value) ret += v;
        return ret;
    }

    static vector<type> evaluate(const vector<type>& value){
        // value[i] := f(source -> child i)
        // we would like to obtain f(child i -> source)
        // child j (j != i) is the grandchildren of child i
        // represent f(child i -> source) using f(source -> j) (j != i)
        // L[i + 1] := the result using f(source -> k) (k = 0, 1, ..., i)
        // R[i] := the result using f(source -> k) (k = i, i + 1, ..., n_children)

        const ll n_children = value.size();
        vl L(n_children + 1, 0), R(n_children + 1, 0);
        rep(i, n_children)
            L[i + 1] = L[i] + value[i];
        revrep(i, n_children)
            R[i] = R[i + 1] + value[i];

        vl ret(n_children);
        rep(i, n_children) ret[i] = L[i] + R[i + 1] + 1;

        return ret;
    }
};

struct StronglyConnectedComponents{
    /*
    dag: [cmp idx][# of connected cmp]
        the index of connected cmp
    cmp: [cmp idx][# of vertex included in a cmp]
        the index of vertex
    cmp_idx: [vertex idx]
        the component which the vertex belongs to

    Use graph_rev when you would like to know the vertex with zero indegree
    */
    ll n, n_cmp;
    vl2 graph, graph_rev, dag, cmp;
    vl order, visited, cmp_idx;

    StronglyConnectedComponents(){}
    StronglyConnectedComponents(ll sz):
        n(sz), graph(sz), graph_rev(sz),
        visited(sz), cmp_idx(sz){}

    void add_edge(ll from, ll to){
        graph[from].pb(to);
        graph_rev[to].pb(from);
    }

    void input(ll m, ll offset = -1){
        ll a, b;
        rep(i, m){
            in2(a, b);
            add_edge(a + offset, b + offset);
        }
    }

    ll operator[](ll k){return cmp_idx[k];}

    void dfs_fwd(ll source){
        visited[source] = 1;
        rep(i, graph[source].size()){
            ll target = graph[source][i];
            if (!visited[target]) dfs_fwd(target);
        }
        order.pb(source);
    }

    void dfs_bwd(ll source, ll num){
        visited[source] = 1, cmp_idx[source] = num;

        cmp[num].pb(source);
        rep(i, graph_rev[source].size()){
            ll target = graph_rev[source][i];
            if (!visited[target]) dfs_bwd(target, num);
        }
    }

    ll build(){
        fill(all(visited), 0);
        order.clear();
        rep(i, n)
            if (!visited[i]) dfs_fwd(i);

        fill(all(visited), 0);
        ll num = 0;
        revrep(i, order.size()){
            if (!visited[order[i]]){
                dag.pb(vl());
                cmp.pb(vl());
                dfs_bwd(order[i], num++);
            }
        }

        rep(i, n) for (ll to: graph[i])
            if (cmp_idx[i] != cmp_idx[to])
                dag[cmp_idx[i]].pb(cmp_idx[to]);

        rep(i, num){
            sort(all(dag[i]));
            dag[i].erase(unique(all(dag[i])), dag[i].end());
        }
        return n_cmp = num;
    }

    // if the v-th vertex is in loop or not
    bool in_loop(ll v){return cmp[cmp_idx[v]].size() > 1;}

    // if the dag has loops or not
    bool has_loop(){
        rep(i, cmp.size()) if (cmp[i].size() > 1) return true;
        return false;
    }
};

struct CombinationMemo{
    ll sz, mod;
    vl fact, fact_inv, minv;

    CombinationMemo(ll sz, ll _mod):
        sz(sz), mod(_mod){
            fact.resize(sz + 5);
            fact_inv.resize(sz + 5);
            minv.resize(sz + 5);
            init();
        }

    void init(){
        fact[0] = fact[1] = 1;
        minv[1] = 1;
        fact_inv[0] = fact_inv[1] = 1;

        For(i, 2, sz + 3){
            fact[i] = (i * fact[i - 1]) % mod;
            minv[i] = mod - minv[mod % i] * (mod / i) % mod;
            fact_inv[i] = fact_inv[i - 1] * minv[i] % mod;
        }
    }

    ll nCk(ll n, ll r){
        if (n == r && n == 0) return 1;
        else if (n <= 0 || r < 0 || r > n) return 0;
        ll val = (fact[n] * fact_inv[n - r]) % mod;
        val *= fact_inv[r];
        return val % mod;
    }

    ll nPk(ll n, ll r){
        if (n == r && n == 0) return 1;
        else if (n <= 0 || r < 0 || r > n) return 0;
        ll val = (fact[n] * fact_inv[n - r]) % mod;
        return val % mod;
    }
};

struct PowerMemo{
    ll sz, mod, base;
    vl powB;

    PowerMemo(ll sz, ll base, ll _mod):
        sz(sz), base(base), mod(_mod){
            powB.resize(sz + 5);
            init();
        }

    void init(){
        powB[0] = 1;
        rep(i, sz + 3)
            powB[i + 1] = (powB[i] * base) % mod;
    }

    ll operator[](ll k){
        return powB[k];
    }
};

struct Grid2D{
    Graph graph;
    ll Width, Height;

    Grid2D(ll w, ll h, vb2& block) : Width(w), Height(h){
        graph.resize(w * h);
        _build(block);
    }

    ll pos_to_idx(point p){
        return p.y * Width + p.x;
    }

    point idx_to_pos(ll idx){
        return {idx % Width, idx / Width};
    }

    bool undefined_region(point p, vb2& block){
        if (p.x < 0 || p.x > Width - 1) return true;
        if (p.y < 0 || p.y > Height - 1) return true;
        if (block[p.x][p.y]) return true;
        return false;
    }

    void _build(vb2& block, ll val = 1){
        rep(x, Width) rep(y, Height){
            point p = {x, y};
            ll idx1 = pos_to_idx(p); ll idx2;

            if (block[x][y]) continue;
            rep(i, 4){
                point nxt = p + dirs[i];
                idx2 = pos_to_idx(nxt);
                if (!undefined_region(nxt, block))
                    graph[idx1].pb({idx2, val}); // dist[idx1][idx2] = val; (warshall-floyd)
            }
        }
    }
};

struct Cumulative2D{
    vl2 cum;
    ll w, h;

    Cumulative2D(ll w, ll h) : w(w), h(h){
        cum = vl2(w + 1, vl(h + 1, 0));
    }

    template<typename T>
    void build(vector<T>& vec){
        // never forget building
        rep(x, w + 1) cum[x][0] = 0;
        rep(y, h + 1) cum[0][y] = 0;
        rep(y, h) rep(x, w)
            cum[x + 1][y + 1] = cum[x][y + 1] + vec[x][y];
        rep(x, w + 1) rep(y, h)
            cum[x][y + 1] += cum[x][y];
    }

    ll func(ll x, ll y, ll dx, ll dy){
        // 1-indexed
        // the rectangle of (x, y), (x + dx, y), (x, y + dy) and (x + dx, y + dy)
        // think about the case of (1, 1, 1, 1).
        if (x + dx > w || y + dy > h) return - INF;
        ll val = cum[x + dx][y + dy];
        val += cum[x][y];
        val -= cum[x][y + dy];
        val -= cum[x + dx][y];
        return val;
    }
};

struct LinearSystemIncidence{
    struct Edge{ll idx, to; bool fwd;};
    /*
    Ax = c
    A: A in R^(V * E) <- given
    x: x in R^E <- estimate this vector
    c: c in R^V <- given
    */
    ll n_vertex, n_edge;
    bool mod2;
    vl x;
    vb visited;
    vector<vector<Edge>> graph;
  
    LinearSystemIncidence(ll n, vpl& es, bool mod2 = false) : 
        n_vertex(n), n_edge(es.size()), mod2(mod2){
        graph.resize(n); visited.resize(n);
        rep(i, n_edge){
            auto& e = es[i];
            graph[e.e1].pb({i, e.e2, true});
            graph[e.e2].pb({i, e.e1, false});
        }
    }
 
    ll dfs(vl& c, ll src){
        visited[src] = true;
        ll ret = c[src];
        for (Edge& e : graph[src]){
            if (visited[e.to]) continue;
            ll tmp = dfs(c, e.to);
            x[e.idx] = e.fwd ? tmp : -tmp;
            ret += tmp;
            if (mod2) x[e.idx] = modf(x[e.idx], 2);
        }
        return mod2 ? ret % 2 : ret;
    }
 
    bool solve(vl& c){
        x.assign(n_edge, 0);
        visited.assign(n_vertex, false);
        rep(src, n_vertex){
            if (visited[src]) continue;
            if (dfs(c, src)) return false;
        }
        return true;
    }
};

struct BigInt10{
    string num;
    map<ll, vl> mod;

    BigInt10(string num = "0") : num(num){}
    size_t size(){return num.size();}

    void compute_mod(ll _mod){
        ll sz = num.size();
        mod[_mod] = vl(sz + 1, 0);
        ll tenth = 1;

        rep(i, sz){
            ll top = num[sz - 1 - i] - '0';
            mod[_mod][i + 1] = (mod[_mod][i] + top * tenth) % _mod;
            (tenth *= 10) %= _mod;
        }
    }

    ll operator[](ll k){return num[k] - '0';}
    ll subseq_mod(ll l, ll r, ll m){
        // s1 s2 ... sl .... sr ... sn
        // the modulo of 00 ... 0 sl ... sr 00 ... 0
        assert(l <= r);
        ll sz = num.size();
        ll tot = mod[m][sz];
        ll ls = modf(tot - mod[m][sz - l], m);
        ll rs = mod[m][sz - r];
        return (tot - ls - rs + 2 * m) % m;
    }

    bool operator<(BigInt10& n2) const{
        if (num.length() < n2.size()) return true;
        else if (num.length() > n2.size()) return false;
        rep(i, num.length()) if (ll(num[i] - '0') != n2[i]) return ll(num[i] - '0') < n2[i];
        return false;
    }

    bool operator<=(BigInt10& n2) const{
        if (num.length() < n2.size()) return true;
        else if (num.length() > n2.size()) return false;
        rep(i, num.length()) if (ll(num[i] - '0') != n2[i]) return ll(num[i] - '0') < n2[i];
        return true;
    }

    bool operator>(BigInt10& n2) const{
        if (num.length() > n2.size()) return true;
        else if (num.length() < n2.size()) return false;
        rep(i, num.length()) if (ll(num[i] - '0') != n2[i]) return ll(num[i] - '0') > n2[i];
        return false;
    }

    bool operator>=(BigInt10& n2) const{
        if (num.length() > n2.size()) return true;
        else if (num.length() < n2.size()) return false;
        rep(i, num.length()) if (ll(num[i] - '0') != n2[i]) return ll(num[i] - '0') > n2[i];
        return true;
    }
};
istream& operator>>(istream& is, BigInt10& a) {return is >> a.num;}
ostream& operator<<(ostream& os, BigInt10& a) {return os << a.num;}

BigInt10 add(BigInt10 s1, BigInt10 s2){
    ll sz1 = s1.size(), sz2 = s2.size(), sz = max(sz1, sz2);
    ll r = abs(sz2 - sz1); string a(r, '0');
    if (sz1 < sz2)
        s1.num = a + s1.num;
    else if (sz1 > sz2)
        s2.num = a + s2.num;
        
    bool inc = false;
    rep(i, sz){
        ll nxt = s1[sz - 1 - i] + s2[sz - 1 - i] + inc;
        if (nxt >= 10) inc = true;
        else inc = false;
        s1.num[sz - 1 - i] = to_string(nxt % 10)[0];
    }
    if (inc) s1.num = "1" + s1.num;
    return s1;
}

BigInt10 add(BigInt10 s, ll x){return add(s, BigInt10(to_string(x)));}

BigInt10 subtract(BigInt10 s1, BigInt10 s2){
    ll sz1 = s1.size(), sz2 = s2.size(), sz = max(sz1, sz2);
    assert(sz1 >= sz2);
    ll r = abs(sz2 - sz1); string a(r, '0');
    if (sz1 > sz2)
        s2.num = a + s2.num;
        
    bool dec = false;
    rep(i, sz){
        ll nxt = s1[sz - 1 - i] - s2[sz - 1 - i] - dec;
        if (nxt < 0) dec = true;
        else dec = false;
        s1.num[sz - 1 - i] = to_string((nxt + 10) % 10)[0];
    }
    ll n0 = 0;
    while (s1.num[n0] == '0') n0++;
    s1.num = substring(s1.num, n0, sz); 
    return s1;
}

BigInt10 subtract(BigInt10 s, ll x){return subtract(s, BigInt10(to_string(x)));}

BigInt10 multiply(BigInt10 s, ll x){
    assert(x <= 10);
    assert(x >= 0);

    if (x == 0) return BigInt10("0");
    else if (x == 1) return BigInt10(s.num);
    if (x == 10) return BigInt10(s.num + "0");

    ll sz = s.size(); ll inc = 0;
    rep(i, sz){
        ll nxt = s[sz - 1 - i] * x + inc;
        inc = nxt / 10;
        s.num[sz - 1 - i] = to_string(nxt % 10)[0];
    }
    if (inc) s.num = to_string(inc) + s.num;
    return s;
}

BigInt10 multiply(BigInt10 s1, BigInt10 s2){
    ll sz = s2.size();
    BigInt10 ret;

    rep(i, sz){
        ll x = s2[sz - 1 - i];
        ret = add(ret, multiply(s1, x));
        s1.num += "0";
    }
    return ret;
}

BigInt10 shift(BigInt10 s, ll k){
    if (k >= s.size()) return BigInt10("0");
    s.num = substring(s.num, 0, s.size() - k - 1);
    return s;
}

struct BigInt2{
    string num;
    ll sz;
    map<ll, vl> mod;
    ll nz;

    BigInt2(string num) : num(num), sz(num.size()){
        nz = 0;
        rep(i, sz) if (num[i] == '1') nz++;
    }
    size_t size(){return num.size();}

    void compute_mod(ll _mod){
        mod[_mod] = vl(sz + 1, 0);
        ll base = 1;

        rep(i, sz){
            ll top = num[sz - 1 - i] - '0';
            mod[_mod][i + 1] = (mod[_mod][i] + top * base) % _mod;
            (base *= 2) %= _mod;
        }
    }

    ll bit_count(){return nz;}
    ll operator[](ll k){return num[k] - '0';}
    ll subseq_mod(ll l, ll r, ll m){
        // s1 s2 ... sl .... sr ... sn
        // the modulo of 00 ... 0 sl ... sr 00 ... 0
        assert(l <= r);
        ll tot = mod[m][sz];
        ll ls = modf(tot - mod[m][sz - l], m);
        ll rs = mod[m][sz - r];
        return modf(tot - ls - rs, m);
    }

    bool operator<(BigInt10& n2) const{
        if (num.length() < n2.size()) return true;
        else if (num.length() > n2.size()) return false;
        rep(i, num.length()) if (ll(num[i] - '0') != n2[i]) return ll(num[i] - '0') < n2[i];
        return false;
    }

    bool operator>(BigInt10& n2) const{
        if (num.length() > n2.size()) return true;
        else if (num.length() < n2.size()) return false;
        rep(i, num.length()) if (ll(num[i] - '0') != n2[i]) return ll(num[i] - '0') > n2[i];
        return num > n2.num;
    }
};
istream& operator>>(istream& is, BigInt2& a) {return is >> a.num;}
ostream& operator<<(ostream& os, BigInt2& a) {return os << a.num;}

struct MedianManager{
    ll sz = 0, median = 0;
    priority_queue<ll> lower;
    priority_queue<ll, vl, greater<ll>> upper;
    vl in, out;
    ll ini;

    MedianManager(ll _ini = -INF){
        ini = _ini;
        in = out = vl(2, _ini);
    }

    void update(ll v){
        sz++;
        if (sz == 1) median = v, in[0] = in[1] = out[0] = out[1] = ini;
        else if (sz % 2 == 0){
            if (v < median) lower.push(v), upper.push(median), in[0] = v, in[1] = median, out[0] = out[1] = ini;
            else lower.push(median), upper.push(v), in[0] = median, in[1] = v, out[0] = out[1] = ini;
        }else if (sz % 2){
            ll lower_top = lower.top(), upper_bottom = upper.top();
            if (v < lower_top) median = lower_top, lower.pop(), lower.push(v), in[0] = v, out[0] = median, in[1] = out[1] = ini;
            else if (v > upper_bottom) median = upper_bottom, upper.pop(), upper.push(v), in[1] = v, out[1] = median, in[0] = out[0] = ini;
            else median = v, in[0] = out[0] = in[1] = out[1] = ini;
        }
    }

    ll size(){return sz;}
    pl medians(){return sz % 2 ? mp(median, median) : mp(lower.top(), upper.top());}
};

struct RangeTypeQuery{
    ll n, cur, Q;
    vl types, most_right, ans;
    vtp3 queries;
    BinaryIndexedTree bit;

    RangeTypeQuery(ll _n, ll _Q, BinaryIndexedTree _bit): bit(_bit){
        n = _n, Q = _Q, cur = 0;
        queries.resize(Q); ans.resize(Q);
        types.resize(n); most_right = vl(n, -1);
    }

    void get_queries(ll offset=-1){
        rep(i, Q){
            ll l, r; in2(l, r);
            l += offset, r += offset;
            queries[i] = {r, l, i};
        }
        sort(all(queries));
    }

    void get_sequence(ll offset=-1){
        rep(i, n){
            in1(types[i]);
            types[i] += offset;
            // type num must be less than n.
            assert(types[i] <= n - 1);
        }
    }

    void update(ll nxt){
        while (cur <= nxt){
            if (most_right[types[cur]] != -1)
                bit.update(most_right[types[cur]], -1);
            most_right[types[cur]] = cur;
            bit.update(most_right[types[cur]], 1);
            cur++;
        }
    }

    void process_queries(){
        rep(i, Q){
            ll l, r, v, idx;
            tie(r, l, idx) = queries[i];
            update(r);
            v = bit.query(l, r);
            ans[idx] = v;
        }
    }
};

ll gcd(ll m, ll n){
    ll a = max(m, n);
    ll b = min(m, n);
    while(b != 1 && b != 0){ a %= b; swap(a, b);}
    return b == 1 ? 1 : a;
}

ll lcm(ll m, ll n){ return m / gcd(m, n) * n;}

ll power_normal(ll a, ll power){
    ll value = 1;
    while(power != 0){
        if(power & 1) value = value * a;
        a = a * a;
        power = power >> 1;
    }
    return value;
}

ll power_mod(ll a, ll power, ll mod){
    __int128_t value = 1;
    __int128_t b = a;
    while(power != 0){
        if (power & 1) value = (value * b) % mod;
        b = (b * b) % mod;
        power = power >> 1;
    }
    return value % mod;
}

ll modinv(ll a, ll mod){
    return power_mod(a, mod - 2, mod);
}

ll combination(ll n, ll r, ll mod){
    if (n == r && n == 0) return 1;
    else if (n <= 0 || r < 0 || r > n) return 0;
    ll numerator = 1;
    ll denomenator = 1;
    for(ll i = 0; i < r ; i++){
        ll num = (n - i) % mod, den = (i + 1) % mod;
        (numerator *= num) %= mod;
        (denomenator *= modinv(den, mod)) %= mod;
    }

    return (numerator * denomenator) % mod;
}

vl2 pascal_triangle(ll n){
    /*
    Complexity: O(n^2)
    The upper bound of n is nearly 50.

    Parameters
    ----------
    n; the size of returned vector

    Returns
    -------
    comb[i][j]: combination(i, j). 0 <= i <= n, 0 <= j <= i
    */

    vl2 comb(n + 1, vl(n + 1));
    comb[0][0] = 1;
    For(i, 1, n + 1) rep(j, i + 1){
        comb[i][j] += comb[i - 1][j];
        if (j > 0) comb[i][j] += comb[i - 1][j - 1];
    }

    return comb;
}

ld log_combination(ll n, ll r){
    if (n == r && n == 0) return 0;
    else if (n <= 0 || r < 0 || r > n) return -INF;
    ld val = 0;
    for(ll i = 0; i < r ; i++){
        val += log(n - i);
        val -= log(i + 1);
    }

    return val;
}

string bin_expression(ll n, ll dig){
    string s = "";
    rep(i, dig){
        s += to_string(n % 2);
        n /= 2;
    }
    reverse(all(s));
    return s;
}

bool is_prime(ll n){
    if (n <= 1) return false;
    for(ll i = 2; i * i <= n; i++){
        if (n % i == 0) return false;
    }
    return true;
}

map<ll, ll> prime_factorization(ll n){
    ll i = 2;
    map<ll, ll> table;
    while(i * i <= n){
        while(n % i == 0){
            table[i]++;
            n /= i;
        }
        i++;
    }
    if (n > 1) table[n] = 1;
    return table;
}

vl divisor_table(ll n, bool asc = true){
    vl table;
    ll i = 1;
    while(i * i <= n){
        if (n % i == 0){
            table.pb(i);
            if (i * i != n) table.pb(n / i);
        }
        i++;
    }

    if (asc) sort(all(table));
    else sort(rall(table));

    return table;
}

ll euler_function(ll x){
    // the number of coprime numbers for x (including 1)
    vl dt = divisor_table(x, false);
    ll ret = x; map<ll, ll> cnt;

    for (ll d: dt){
        if (d == 1) continue;
        cnt[d] = x / d;
        for (auto& c: cnt)
            if (c.e1 != d && c.e1 % d == 0)
                cnt[d] -= c.e2;
        ret -= cnt[d];
    }
    return ret;
}

vl base_converter(ll num, ll base, ll sz){
    vl coefs(sz); ll cur = 0;
    while (num){
        ll val = num % base;
        assert(cur < sz);
        coefs[cur++] = val;
        num -= val, num /= base;
    }
    while (cur < sz) coefs[cur++] = 0;
    return coefs;
}

string indexed_alphabet(ll leng, ll idx){
    // 0-indexed
    // returns the string whose order is the-idx-th in the set of strings of the given length.
    string ret = "";
    if (idx >= power_normal(26, leng))
        return ret;

    vl cs = base_converter(idx, 26, leng);
    reverse(all(cs));
    rep(i, cs.size()) ret += char('a' + cs[i]);

    return ret;
}

ll alphabetical_index(string s){
    // including the strings whose length is less than that of s.
    // excluding the empty string.
    ll leng = s.length();
    ll ret = 0, p = 1;
    rep(i, leng){
        ll b = s[leng - 1 - i] - 'a';
        ret += p + p * b, p *= 26;
    }
    return ret;
}

// just change the input if you want to change the target.
// If you want to check the common sequences in two strings,
// combine them. e.g. ABC150F
template<typename T>
vl z_algorithm(T& s, ll n){
    /*
    Paramters
    ---------
    T: the string or list of interest
    n: the size of string or list

    Returns
    -------
    res[i] is the maximum number of K which satisfies
    s[:K] == s[i:i + K]
    for each i = 0, 1, 2, ..., n - 1.
    */
    vl res(n);
    res[0] = n;
    ll i1 = 1, i2 = 0;

    while(i1 < n){
        /*
        i1: the starting point
        i2: the length of substring
        */
        while (i1 + i2 < n && s[i2] == s[i1 + i2]) ++i2;
        res[i1] = i2; if (i2 == 0){++i1; continue;}
        ll i3 = 1;

        // update the already seen points
        while (i1 + i3 < n && i3 + res[i3] < i2){res[i1 + i3] = res[i3]; ++i3;}
        // update up to i1 + i3 and the next possible minimum length is i2 - i3 (= res[i3])
        i1 += i3, i2 -= i3;
    }
    return res;
}

ll string_to_ll(string s){
    ll l = s.length(); ll idx = 0;
    ll val = 0; ll tenth = 1;
    while (idx < l){
        ll m = s[l - 1 - idx] - '0';
        val += (m * tenth);
        tenth *= 10; idx++;
    }

    return val;
}

string reflected_string(string s){
    string t, u; ll n = s.length();
    t = s; reverse(all(t));
    u = substring(t, 0, n - 2) + s + substring(t, 1, n - 1);
    return u;
}

ld distance_between_point_line(point l_begin, point l_end, point p){
    ll xl1 = l_begin.x, yl1 = l_begin.y;
    ll xl2 = l_end.x, yl2 = l_end.y;
    ll xp = p.x, yp = p.y;
    ll a = yl2 - yl1;
    ll b = - xl2 + xl1;
    ll c = - a * xl2 - b * yl2;
    return abs(ld(a * xp + b * yp + c)) / ld(sqrt(a * a + b * b));
}

bool is_cross(point l1_begin, point l1_end, point l2_begin, point l2_end){
    ll x1 = l1_begin.x, y1 = l1_begin.y;
    ll x2 = l1_end.x, y2 = l1_end.y;
    ll x3 = l2_begin.x, y3 = l2_begin.y;
    ll x4 = l2_end.x, y4 = l2_end.y;
    ll val1 = (x1 - x2) * (y3 - y1) + (y1 - y2) * (x1 - x3);
    ll val2 = (x1 - x2) * (y4 - y1) + (y1 - y2) * (x1 - x4);
    ll val3 = (x3 - x4) * (y1 - y3) + (y3 - y4) * (x3 - x1);
    ll val4 = (x3 - x4) * (y2 - y3) + (y3 - y4) * (x3 - x2);

    return val1 * val2 < 0 && val3 * val4 < 0;
}

template<typename T>
bool isColinear(T p1, T p2, T p3){
    T v1 = p2 - p1, v2 = p3 - p1;
    return v1.x * v2.y == v1.y * v2.x;
}

template<typename T>
T PerpendicularBisector(T p1, T p2){
    T vec = p2 - p1;
    assert(!vec.is_zero());
    T ret = {vec.y, -vec.x};

    return ret;
}

template<typename T>
ld Distance2DPoints(T p1, T p2){
    T vec = (p1 - p2) * (p1 - p2);
    return sqrt(vec.x + vec.y);
}

ll SquaredDistance2DPoints(point p1, point p2){
    point vec = (p1 - p2) * (p1 - p2);
    return vec.x + vec.y;
}

ld space_of_triangle(point p1, point p2, point p3){
    ll x1 = p1.x, y1 = p1.y;
    ll x2 = p2.x, y2 = p2.y;
    ll x3 = p3.x, y3 = p3.y;
    ll v1 = x2 - x1; ll u1 = y2 - y1;
    ll v2 = x3 - x1; ll u2 = y3 - y1;
    ld s = ld(v1 * u2 - u1 * v2) / ld(2);
    return abs(s);
}

pair<point, ll> OuterCenter(point p1, point p2, point p3){
    // the center of circle on the given three points
    // return the determinant value and the product of center points and 2 * determinant value
    point ret;

    if (isColinear(p1, p2, p3)){
        ll d1 = SquaredDistance2DPoints(p1, p2);
        ll d2 = SquaredDistance2DPoints(p2, p3);
        ll d3 = SquaredDistance2DPoints(p3, p1);
        if (d1 >= d2 && d1 >= d3) {ret = p1 + p2; return mp(ret, 2);}
        else if (d2 >= d1 && d2 >= d3) {ret = p2 + p3; return mp(ret, 2);}
        else {ret = p3 + p1; return mp(ret, 2);}
    }

    point pv1 = PerpendicularBisector(p1, p2);
    point pv2 = PerpendicularBisector(p1, p3);
    point cv1_2x = p1 + p2, cv2_2x = p1 + p3;
    // cv1 + k pv1 == cv2 + m pv2
    // (pv1x -pv2x) (k) = (cv2x - cv1x)
    // (pv1y -pv2y) (m) = (cv2y - cv1y)
    ll det_inv = - pv1.x * pv2.y + pv1.y * pv2.x;
    ll x1_2x = cv2_2x.x - cv1_2x.x, x2_2x = cv2_2x.y - cv1_2x.y;
    pl c_2x_det = {-pv2.y * x1_2x + pv2.x * x2_2x, -pv1.y * x1_2x + pv1.x * x2_2x};
    
    // ret.x = ld(cv1_2x.x * det_inv + pv1.x * c_2x_det.e1) / ld(2 * det_inv);
    // ret.y = ld(cv1_2x.y * det_inv + pv1.y * c_2x_det.e1) / ld(2 * det_inv);
    ret.x = cv1_2x.x * det_inv + pv1.x * c_2x_det.e1;
    ret.y = cv1_2x.y * det_inv + pv1.y * c_2x_det.e1;

    ll jacobian = 2 * det_inv;
    return mp(ret, jacobian);
}

ll inversion_number(vl a, ll a_max){
    /*
    Paramters
    ---------
    a: vector<ll>
        All the elements must be non-negative.
        Prefably the elements are compressed to reduce the computational cost.
    a_max: ll
        The maximum value of the vector a or the value bigger than the value stated previously.
    */

    BinaryIndexedTree bit(a_max + 1); ll val = 0;
    rep(i, a.size()){
        // i is the number of elements that have lower index than a[i].
        // call the number of elements that have lower value than a[i]
        // by subtracting these two, the residual number is the number of elements that have larger value.
        val += i - bit.query(a[i] - 1);// cumulative sum from 0 to a[i] - 1
        bit.update(a[i], 1);
    }
    return val;
}

ld bin_search(ld left, ld right, bool lb, function<bool(ld)> judge){
    ld mid;
    while(right - left > EPS * 5){
        mid = (right + left) / 2;
        if (lb){
            if (judge(mid)) right = mid;
            else left = mid + EPS;
        }else{
            if (judge(mid)) left = mid;
            else right = mid - EPS;
        }
    }
    return right;
}

ll bin_search(ll left, ll right, bool lb, function<bool(ll)> judge){
    ll mid;
    while(right > left){
        if (lb){
            // if true (satisfies the condition), range shifts smaller direction
            mid = (right + left) / 2;
            if (judge(mid)) right = mid;
            else left = mid + 1;
        }else{
            // if true (satisfies the condition), range shitfs larger direction
            mid = (right + left + 1) / 2;
            if (judge(mid)) left = mid;
            else right = mid - 1;
        }
    }
    return right;
}

ld trinary_search(ld left, ld right, function<ld(ld)> func){
    // Care the value EPS!!! Compare to the condition
    while(abs(right - left) > EPS){
        ld left2 = (2.0 * left + right) / 3.0;
        ld right2 = (left + 2.0 * right) / 3.0;
        ld f1 = func(left2);
        ld f2 = func(right2);
        if (f1 <= f2) right = right2;
        else if (f2 <= f1) left = left2;
    }

    return right;
}

template<typename T>
vector<T> compress(vector<T> v){
    // sort and remove all the duplicated values
    sort(all(v));
    v.erase(unique(all(v)), v.end());
    return v;
}

template<typename T>
map<T, ll> dict(const vector<T> &v){
    map<T, ll> d;
    rep(i, v.size()) d[v[i]] = i;
    return d;
}

points compress2D(vl xs, vl ys){
    /*
    NOTE
    ----
    Add the corner points if required
    */

    ll n = xs.size();
    vl xcs = compress(xs), ycs = compress(ys);
    map<ll, ll> xd = dict(xcs), yd = dict(ycs);
    points ps(n);

    rep(i, n)
        xs[i] = xd[xs[i]], ys[i] = yd[ys[i]];

    rep(i, n)
        ps[i] = {xs[i], ys[i]};

    sort(all(ps));

    return ps;
}

ll kruskal(vector<undirected_edge>& es, ll n_vertex){
    // O(ElogE)
    sort(all(es));
    UnionFind uf(n_vertex);

    ll min_cost = 0;

    rep(i, es.size()){
        undirected_edge& e = es[i];
        if (!uf.is_same(e.from, e.to)){
            min_cost += e.cost;
            uf.unite(e.from, e.to);
        }
    }
    return min_cost;
}

ll LongestIncreasedSequence(vl& v){
    ll n = v.size();
    vl dp(n, INF);
    rep(i, n)
        *lower_bound(all(dp), v[i]) = v[i];
    return lower_bound(all(dp), INF) - dp.begin();
}

void dijkstra(ll start, Graph& graph, vl& dist,
              vl& vertex_pre, bool trace = false){
    priority_queue<pl, vpl, greater<pl>> q;
    ll n = graph.size(); dist = vl(n, INF); dist[start] = 0;
    if (trace) vertex_pre = vl(n, -1);
    q.push(pl(0, start));

    while (!q.empty()){
        ll idx, cost;
        tie(cost, idx) = q.top(); q.pop();
        if (dist[idx] < cost) continue;

        for (auto& e: graph[idx]) 
            if (chmin(dist[e.to], dist[idx] + e.cost)){
                if (trace) vertex_pre[e.to] = idx;
                q.push(pl(dist[e.to], e.to));
            }
    }
}

vl get_predecessor(ll g, vl& vertex_pre){
    vl path;
    for (; g != -1; g = vertex_pre[g]) path.pb(g);

    reverse(all(path));
    return path;
}

void warshall_floyd(vl2& dist){
    ll n = dist.size();
    // Dont forget the initialization
    // rep(i, n) rep(j, n) dist[i][j] = INF * (i != j);
    rep(k, n) rep(i, n) rep(j, n) dist[i][j] = min(dist[i][j], dist[i][k] + dist[k][j]);
}

void update_warshall_floyd(vl2& dist, ll v1, ll v2, ll cost){
    ll n = dist.size();
    if (!chmin(dist[v1][v2], cost)) return;
    dist[v2][v1] = dist[v1][v2];
    rep(i, n) rep(j, n) dist[i][j] = min(dist[i][j], dist[i][v1] + dist[v1][j]);
    rep(i, n) rep(j, n) dist[i][j] = min(dist[i][j], dist[i][v2] + dist[v2][j]);
}

// ABC061D, 137E
bool find_negative_cycle(ll n, ll goal, Graph& graph, vl& dist){
    rep(i, n) rep(v, n) rep(k, graph[v].size()){
        edge e = graph[v][k];
        if (dist[e.to] != INF && dist[e.to] > dist[v] + e.cost){
            dist[e.to] = -INF;
            if (goal == -1) return true;
            else if (goal == e.to) return true;
        }
    }
    return false;
}

bool bellman_ford(ll start, ll goal, Graph& graph, vl& dist){
    // if there is a closed circuit, it returns false. (when goal == -1)
    // if the distance to goal cannot be obtained, it returns false (when goal != -1)
    ll n = graph.size();
    dist = vl(n, INF);
    dist[start] = 0;
    rep(i, n) rep(v, n) rep(k, graph[v].size()){
        edge e = graph[v][k];
        if (dist[v] != INF && dist[e.to] > dist[v] + e.cost)
            dist[e.to] = dist[v] + e.cost;
    }

    if (find_negative_cycle(n, goal, graph, dist)) return false;
    return true;
}

void Euler_Tour(Graph& tree, vl& euler_tour, vl& in, vl& out, ll anc = 0){
    // record only when we first reach a node and leave the node.
    ll n = tree.size();
    vb visited(n, false);
    in = out = vl(n);

    auto dfs = [&](auto dfs, ll source) -> void{
        visited[source] = true;
        in[source] = euler_tour.size();
        euler_tour.pb(source);

        bool is_leaf = true;
        for (auto& e: tree[source]){
            ll target = e.to;
            if (visited[target]) continue;
            else is_leaf = false;
            dfs(dfs, target);
        }
        
        euler_tour.pb(- source);
        out[source] = euler_tour.size() - 1;
    };

    dfs(dfs, anc);
}

void Euler_Tour2(Graph& tree, vl& euler_tour, vl& in, vl& out, ll anc = 0){
    // record everytime we reach a node
    ll n = tree.size();
    vb visited(n, false);
    in = out = vl(n);

    auto dfs = [&](auto dfs, ll source) -> void{
        visited[source] = true;
        in[source] = euler_tour.size();
        euler_tour.pb(source);

        bool is_leaf = true;
        for (auto& e: tree[source]){
            ll target = e.to;
            if (visited[target]) continue;
            else is_leaf = false;
            dfs(dfs, target);
            euler_tour.pb(source);
        }

        out[source] = euler_tour.size() - 1;
    };

    dfs(dfs, anc);
}

std::mt19937 create_rand_engine(){
    std::random_device rnd;
    std::vector<std::uint_least32_t> v(10);
    std::generate(v.begin(), v.end(), std::ref(rnd));
    std::seed_seq seed(v.begin(), v.end());
    return std::mt19937(seed);
}

vl generate_unique_array(const size_t sz, ll vm, ll vM){
    const size_t range = static_cast<size_t>(vM - vm + 1);
    const size_t num = static_cast<size_t>(sz * 1.2);
    assert(vm <= vM);
    assert(sz <= range);

    vl ret;
    auto engine = create_rand_engine();
    std::uniform_int_distribution<ll> distribution(vm, vM);

    while (ret.size() < sz){
        while (ret.size() < num) ret.pb(distribution(engine));
        sort(all(ret));

        auto unique_end = unique(all(ret));
        if (sz < distance(ret.begin(), unique_end)) 
            unique_end = next(ret.begin(), sz);

        ret.erase(unique_end, ret.end());
    }

    return ret;
}

vl generate_array(const size_t sz, ll vm, ll vM){
    const size_t range = static_cast<size_t>(vM - vm + 1);
    assert(vm <= vM);

    vl ret;
    auto engine = create_rand_engine();
    std::uniform_int_distribution<ll> distribution(vm, vM);

    while (ret.size() < sz)
        ret.pb(distribution(engine));

    return ret;
}

template<typename T>
void read_vector(ll n, vector<T>& v, ll offset = 0){
    v.resize(n);
    rep(i, n){
        in1(v[i]);
        if (offset) v[i] += offset;
    }
}

template<typename T>
void read_vector(ll n, vector<T>& v1, vector<T>& v2, ll offset1 = 0, ll offset2 = 0){
    v1.resize(n); v2.resize(n);
    rep(i, n){
        in2(v1[i], v2[i]);
        if (offset1) v1[i] += offset1;
        if (offset2) v2[i] += offset2;
    }
}

void read_points(ll n, points& ps, ll offset = 0){
    ps.resize(n);
    rep(i, n){
        ll x, y; in2(x, y);
        ps[i] = {x, y};
        ps[i] += offset;
    }
}

void read_2DMap(ll w, ll h, vb2& block, char b){
    block = vb2(w, vb(h, false));
    string s;
    rep(y, h){
        in1(s);
        rep(x, w) block[x][y] = (s[x] == b);
    }
}

ll diameter_of_tree(Graph& tree){
    ll dM = 0, vM = 0, v1, v2;

    auto dfs = [&](auto dfs, ll src, ll par, ll d) -> void{
        if (chmax(dM, d)) vM = src;
        for (auto& e: tree[src]){
            if (e.to == par) continue;
            dfs(dfs, e.to, src, d + 1);
        }
    };
    dfs(dfs, 0, -1, 0); v1 = vM;
    dfs(dfs, vM, -1, 0); v2 = vM;
    return dM;
}

void trit_search(ll n, function<void(ll, ll)> func){
    rep(bit, 1LL << n){
        ll sub = bit;
        do{
            func(bit, sub);
            sub = (sub - 1) & bit;
        }while (sub != bit);
    }
}

ll convert_string_to_int(string s, ll n_below_decimal){
    // return double(s) * coef
    ll ret = 0, comma = s.length(), n10 = 1, coef = power_normal(10, n_below_decimal);
    rep(i, s.length()) if (s[i] == '.') comma = i;
    revrep(i, comma){
        ll num = s[i] - '0';
        ret += n10 * num;
        n10 *= 10;
    }
    ret *= coef; n10 = coef / 10;
    For(i, comma + 1, s.length()){
        ll num = s[i] - '0';
        ret += n10 * num;
        n10 /= 10;
    }
    return ret;
}

void iter_combination(ll n, ll k, function<void(ll)> func){
    /*
        sub & -sub: the binary which shares the last digit whose value is 1 in sub
        sub + x   : carry up the last digit
            ~y    : the binary whose digits are 1 if y's digit is 0.
    (sub & ~y) / x: reduce the same number of 0s after first 1 in x from (sub & ~y).
    */

    auto next_combination = [&](ll sub) -> ll{
        ll x = sub & -sub, y = sub + x;
        if (x != 0) return (((sub & ~y) / x) >> 1) | y;
        else return INF;
    };

    ll M = 1LL << n;
    for (ll bit = (1LL << k) - 1; bit < M; bit = next_combination(bit))
        func(bit);
}

ll count_in_sorted_seq(vl& as, ll val){
    ll lb = lower_bound(all(as), val) - as.begin();
    ll ub = upper_bound(all(as), val) - as.begin();
    return ub - lb;
}

bool win_first_player(vl& grundy_nums){
    ll x = 0;
    for (ll& gn: grundy_nums) x ^= gn;
    return x != 0;
}

ll create1001001(ll num, ll m, ll num_of_1, ll nb = 31){
    // e.g. 1001001001 -> num = 3
    // e.g. 111111 -> num = 1 (REPUNIT)
    // Calculate the modulo m of 100..100...100..1 (the length of each 00... is num - 1. The # of 1 is 2^i.)
    // Then calculate the modulo m of 100...100...100...1 (The # of 1 is num_of_1 using the memo.)
    vl memo(nb, 1); ll v = 0, tot = 1, coef = power_mod(10, num, m);
    rep(j, nb){
        ll c10 = power_mod(coef, 1LL << j, m);
        memo[j + 1] = ((memo[j] * c10) % m + memo[j]) % m;
        if (num_of_1 >> j & 1){
            (v += (memo[j] * tot) % m) %= m;
            (tot *= power_mod(coef, 1LL << j, m)) %= m;
        }
    }
    return v;
}

ll GaussJordanBitVector(vl &bs){
    ll n = bs.size();
    ll rank = 0; ll j = 0;
    revrep(i, N_DIGITS){
        for(j = rank; j < n; j++) if (bs[j] & (1LL << i)) break;
        if (j == n) continue;
        if (j > rank) bs[rank] ^= bs[j];
        for (j = 0; j < n; j++) if (j != rank) bs[j] = min(bs[j], bs[j] ^ bs[rank]);
        rank++;
    }
    return rank;
}

ll GaussJordan(vl2& A, ll mod, bool is_extended = false, bool det = false){
    /*
    Ax = b
    
    Parameters
    ----------
    A: m x n matrix
        The matrix of interest

    is_extended: bool
        if A includes the term of b (e.g. when solving linear equation), true
        otherwise false.

    Property: The product of diagonal elements of upper triangle matrix is equal to the determinant of this matrix.
    */

    ll rank = 0, m = A.size(), n = A[0].size();
    rep(r, m) rep(c, n) A[r][c] = modf(A[r][c], mod);

    ll pivot, inv, ret = 1; vl tmp;
    if (det) tmp = vl(n);
    rep(c, n){
        if (is_extended && c == n - 1) break;
        pivot = -1;
        For(r, rank, m) if (A[r][c] != 0) {pivot = r; break;}
        if (pivot == -1){
            if (det) return 0;
            continue;
        }

        swap(A[pivot], A[rank]);
        ret = modf(ret * A[rank][c], mod), inv =  modinv(A[rank][c], mod);
        vl& ax = det ? tmp : A[rank];
        rep(c2, n) ax[c2] = A[rank][c2] * inv % mod;

        rep(r, m) if (r != rank && A[r][c]){
            ll base = A[r][c];
            rep(c2, n)
                A[r][c2] = modf(A[r][c2] - ax[c2] * base, mod);
        }
        ++rank;
    }

    return det ? ret : rank;
}

ll linear_equation(vl2 A, vl b, vl &res){
    ll m = A.size(), n = A[0].size();
    vl2 M(m, vl(n + 1));
    rep(r, m){
        rep(c, n) M[r][c] = A[r][c];
        M[r][n] = b[r];
    }

    ll rank = GaussJordan(M, true);
    For(r, rank, m) if (M[r][n]) return -1;

    res.assign(n, 0);
    rep(i, rank) res[i] = M[i][n];
    return rank;
}

ll determinant(vl2& A, ll mod){return GaussJordan(A, mod, false, true);};

/*
    #5. shakutori method (syakutori, two pointers technique)
    // 1. strech right side while the condition is met.
    // 2. renew the answer
    // 3. increments left side
    // 4. Back to 1. (l <= r must be satisfied all the time.)

    ll l = 0, r = 0;
    while (l < n){
        if (l == r) r++;
        while(r < n && cond) r++;
        l++;
    }
    prl(answer);

    #11. bfs ABC146D, ABC007C
    1. first create a graph.
    2. start searching from a node.
    3. do some processes and push nodes connected with a given target node in BFS.
    4. repeat a series of procedure until queue is empty.

    ll n; in1(n);
    queue<tp3> q; vb visited(n, false);

    auto bfs = [&](ll src, ll par){
        visited[src] = true;
        for (auto& e: graph[src]){
            if (e.to != par && !visited[e.to]){
                q.push(mt(e.to, src, ...));
            }
        }
    };

    q.push(mt(0, -1, 0));
    while(!q.empty()){
        ll src, par, ...; tie(src, par, ...) = q.front(); q.pop();
        bfs(src, par, ...);
    }

    Grundy number (ARC038C)
    void grundy(ll x, vl& gn){ // the example from the ari book
        // the player can take a[0], a[1], ..., a[k - 1] coins from 1 mountain
        set<ll> s;
        rep(i, k) if (a[i] <= x) s.insert(gn[x - a[i]])
        
        while (s.count(gn[x])) gn[x]++;
    }
*/

/*
      x: 1001 1010  1100 1011 1101  1111
x & ~ x: 0001 0010  0100 0001 0001  0001
    sum: 1010 1100 10000 1100 1100 10000

####### Attention #######
S | (1 << i) -> S union {i}
S & ~(1 << i) -> S - {i}

# inclusion-exclusion principle
|U[i = 1 to n] Ai| = sum[i = 1 to n] |Ai| - sum[i < j]|Ai ^ Aj| + ... + (-1)^(n - 1) |^[i = 1 to n]Ai|
*/

const ll MAX_N = 200005;
const size_t MAX_BIT = 160 * 160 + 10;
typedef bitset<MAX_BIT> bts;
typedef vector<bts> vbt;
typedef vector<vbt> vbt2;
typedef vector<vbt2> vbt3;
const bool multiple_queries = false;

void solve(){
    /*
    *** 注意 ***
    ・マイナスの整数の除算が0 (マイナスのときはCeil関数になる)になって，マイナスの判定ができていない．
    ・MODの逆元を取るときは素数か否かに注意．
    ・Queue, stack, deque等で必ず`empty`なら飛ばすようにする．
    ・set, multisetで`RE`を起こすときは初期段階で解法に影響が出ないようなダミーをsetに入れておくとエラーを回避できる．
    ・掛け算のOverflow注意．(x * y > 10^18　-> x > 10^18 / y) のような形で判定するとよい．
    ・総和を取る公式 (1+2+...+n = n * (n + 1) / 2)の計算時等で最終的には10^18に収まる場合は`/2`の順序を早める．
    ・Bit演算をするときに`a ^ b != c`のような式は代入演算子になったりするので注意． (ABC150F)
    ・aborted (core dumped) のエラーが出ているときは変数の二重宣言やベクトルの区間外参照を確認．
    ・`tuple`のコンパイルエラーは`make_tuple(a, b, c, ...)`で回避．
    ・グラフの最短経路と木の最短経路は同じDFSでも更新方法に注意. (ABC148E)
    ・Running Errorの１つとして，関数の返り値のタイプが定義されているのに何も返していないことが考えられる．
    ・Running Errorとしてベクトルサイズが0であるにもかかわらずsortをしている場合がある．
    ・境界条件は必ず試す．
    */
    // generate_array(size, min, max);

    ll n; in1(n);
    vl as, cnt(n + 1, 0); read_vector(n, as, 1);

    mint ans = 1; cnt[0] = 3;
    rep(i, n){
        ans *= cnt[as[i] - 1] - cnt[as[i]];
        cnt[as[i]]++;
    }
    prl(ans);

    // ### DEBUG PART ###
    auto naive_solver = [&](){
    };
    #ifdef _LOCAL
        naive_solver();
    #endif
}

void test(ll num = 0, bool verbose = false){
    rep(i, max(1LL, num)){
        ll t = clock();
        if (verbose) prl4("\n#####", i + 1, "#####", "\n## Answer ##");
        solve();
        if (verbose) {prl(""); prl_time(t);}
    }
}

int main(void){
    ios::sync_with_stdio(false);
    cin.tie(nullptr);
    #ifdef _LOCAL
        test(6, true);
    #else
        ll t = 0;
        if (multiple_queries) in1(t);
        test(t, false);
    #endif

    return 0;
}