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#include <bits/stdc++.h>
using namespace std;
template <class T>
vector<pair<pair<T, T>, T>> quotient_range(T n) {
//[l,r]:quotient=z
//[l,r]:increasing
// z:decreasing
// z=0 is not included
T m = 1;
vector<pair<pair<T, T>, T>> ret;
while (m * m <= n) {
ret.emplace_back(make_pair(m, m), n / m);
m++;
}
for (T i = m; i >= 1; i--) {
T L = n / (i + 1) + 1;
T R = n / i;
if (L <= R && ret.back().first.second < L) ret.emplace_back(make_pair(L, R), n / L);
}
return ret;
}
template <class T>
vector<pair<pair<T, T>, T>> quotient_range_ceil(T n) {
const T inf = numeric_limits<T>::max();
vector<pair<pair<T, T>, T>> res;
if (n == 1) {
res.push_back({{1, inf}, 1});
return res;
}
T m = 1;
while (m * m < n) {
res.push_back({{m, m + 1}, (n + m - 1) / m});
m++;
}
m = res.back().second - 1;
while (m >= 1) {
T l = (n + m - 1) / m;
T r = m > 1 ? (n + m - 2) / (m - 1) : inf;
if (l < r) res.push_back({{l, r}, m});
m--;
}
return res;
}
#line 1 "Quotient-Range.cpp"
#include <bits/stdc++.h>
using namespace std;
template <class T>
vector<pair<pair<T, T>, T>> quotient_range(T n) {
//[l,r]:quotient=z
//[l,r]:increasing
// z:decreasing
// z=0 is not included
T m = 1;
vector<pair<pair<T, T>, T>> ret;
while (m * m <= n) {
ret.emplace_back(make_pair(m, m), n / m);
m++;
}
for (T i = m; i >= 1; i--) {
T L = n / (i + 1) + 1;
T R = n / i;
if (L <= R && ret.back().first.second < L) ret.emplace_back(make_pair(L, R), n / L);
}
return ret;
}
template <class T>
vector<pair<pair<T, T>, T>> quotient_range_ceil(T n) {
const T inf = numeric_limits<T>::max();
vector<pair<pair<T, T>, T>> res;
if (n == 1) {
res.push_back({{1, inf}, 1});
return res;
}
T m = 1;
while (m * m < n) {
res.push_back({{m, m + 1}, (n + m - 1) / m});
m++;
}
m = res.back().second - 1;
while (m >= 1) {
T l = (n + m - 1) / m;
T r = m > 1 ? (n + m - 2) / (m - 1) : inf;
if (l < r) res.push_back({{l, r}, m});
m--;
}
return res;
}