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#include <bits/stdc++.h>
using namespace std;
using ll = long long;
using pii = pair<int, int>;
using vpii = vector<pii>;
using vi = vector<int>;
using vll = vector<long long>;
using mpii = map<int, int>;
using mpll = map<ll, ll>;
using db = long double;
#define pb push_back
#define all(x) (x).begin(), (x).end()
#define rall(x) (x).rbegin(), (x).rend()
#define lb lower_bound
#define ub upper_bound
#define make_unique(x) \
sort(all((x))); \
(x).resize(unique(all((x))) - (x).begin())
#define ceil(a, b) ((a) + (b)-1) / (b)
const int mod = (int)1e9 + 7;
const db pi = acos((db)-1);
const int dx[4]{1, 0, -1, 0};
const int dy[4]{0, 1, 0, -1};
template <typename k, typename v>
ostream &operator<<(ostream &os, const pair<k, v> &p);
template <typename t> ostream &operator<<(ostream &os, const vector<t> &vec);
template <typename k, typename v>
ostream &operator<<(ostream &os, const map<k, v> &m);
template <typename k, typename v>
ostream &operator<<(ostream &os, const unordered_map<k, v> &m);
template <typename t> ostream &operator<<(ostream &os, const set<t> &s);
template <typename t>
ostream &operator<<(ostream &os, const unordered_set<t> &s);
template <typename k, typename v>
ostream &operator<<(ostream &os, const pair<k, v> &p) {
os << "(" << p.first << ", " << p.second << ")";
return os;
}
template <typename t> ostream &operator<<(ostream &os, const vector<t> &vec) {
os << "[";
for (size_t i = 0; i < vec.size(); ++i) {
if (i > 0)
os << ", ";
os << vec[i];
}
os << "]";
return os;
}
template <typename t>
ostream &operator<<(ostream &os, const vector<vector<t>> &vec) {
os << "[\n";
for (size_t i = 0; i < vec.size(); ++i) {
os << " " << vec[i];
if (i > 0)
os << ", ";
os << '\n';
}
os << "]";
return os;
}
template <typename k, typename v>
ostream &operator<<(ostream &os, const map<k, v> &m) {
os << "{";
for (const auto &p : m) {
os << p.first << ": " << p.second << ", ";
}
os << "}";
return os;
}
template <typename k, typename v>
ostream &operator<<(ostream &os, const unordered_map<k, v> &m) {
os << "{";
for (const auto &p : m) {
os << p.first << ": " << p.second << ", ";
}
os << "}";
return os;
}
template <typename t> ostream &operator<<(ostream &os, const set<t> &s) {
int i = 0;
os << "{";
for (const auto &e : s) {
if (i > 0)
os << ", ";
os << e;
i++;
}
os << "}";
return os;
}
template <typename t>
ostream &operator<<(ostream &os, const unordered_set<t> &s) {
int i = 0;
os << "{";
for (const auto &e : s) {
if (i > 0)
os << ", ";
os << e;
i++;
}
os << "}";
return os;
}
void debug_out() { cerr << endl; }
template <typename head, typename... tail> void debug_out(head h, tail... t) {
cerr << h << '\n';
debug_out(t...);
}
#define dbg(...) cerr << "(" << #__VA_ARGS__ << "):\n", debug_out(__VA_ARGS__)
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