#include <iostream>
#include <string>
#include <vector>
#include <array>
#include <tuple>
#include <stack>
#include <queue>
#include <deque>
#include <algorithm>
#include <set>
#include <map>
#include <unordered_set>
#include <unordered_map>
#include <bitset>
#include <cmath>
#include <functional>
#include <cassert>
#include <climits>
#include <iomanip>
#include <numeric>
#include <memory>
#include <random>
#include <thread>
#include <chrono>
#define allof(obj) (obj).begin(), (obj).end()
#define range(i, l, r) for(int i=l;i<r;i++)
#define unique_elem(obj) obj.erase(std::unique(allof(obj)), obj.end())
#define bit_subset(i, S) for(int i=S, zero_cnt=0;(zero_cnt+=i==S)<2;i=(i-1)&S)
#define bit_kpop(i, n, k) for(int i=(1<<k)-1,x_bit,y_bit;i<(1<<n);x_bit=(i&-i),y_bit=i+x_bit,i=(!i?(1<<n):((i&~y_bit)/x_bit>>1)|y_bit))
#define bit_kth(i, k) ((i >> k)&1)
#define bit_highest(i) (i?63-__builtin_clzll(i):-1)
#define bit_lowest(i) (i?__builtin_ctzll(i):-1)
#define sleepms(t) std::this_thread::sleep_for(std::chrono::milliseconds(t))
using ll = long long;
using ld = long double;
using ul = uint64_t;
using pi = std::pair<int, int>;
using pl = std::pair<ll, ll>;
using namespace std;

template<typename F, typename S>
std::ostream &operator<<(std::ostream &dest, const std::pair<F, S> &p){
  dest << p.first << ' ' << p.second;
  return dest;
}
template<typename T>
std::ostream &operator<<(std::ostream &dest, const std::vector<std::vector<T>> &v){
  int sz = v.size();
  if(sz==0) return dest;
  for(int i=0;i<sz;i++){
    int m = v[i].size();
    for(int j=0;j<m;j++) dest << v[i][j] << (i!=sz-1&&j==m-1?'\n':' ');
  }
  return dest;
}
template<typename T>
std::ostream &operator<<(std::ostream &dest, const std::vector<T> &v){
  int sz = v.size();
  if(sz==0) return dest;
  for(int i=0;i<sz-1;i++) dest << v[i] << ' ';
  dest << v[sz-1];
  return dest;
}
template<typename T, size_t sz>
std::ostream &operator<<(std::ostream &dest, const std::array<T, sz> &v){
  if(sz==0) return dest;
  for(int i=0;i<sz-1;i++) dest << v[i] << ' ';
  dest << v[sz-1];
  return dest;
}
template<typename T>
std::ostream &operator<<(std::ostream &dest, const std::set<T> &v){
  for(auto itr=v.begin();itr!=v.end();){
    dest << *itr;
    itr++;
    if(itr!=v.end()) dest << ' ';
  }
  return dest;
}
template<typename T, typename E>
std::ostream &operator<<(std::ostream &dest, const std::map<T, E> &v){
  for(auto itr=v.begin();itr!=v.end();){
    dest << '(' << itr->first << ", " << itr->second << ')';
    itr++;
    if(itr!=v.end()) dest << '\n';
  }
  return dest;
}
std::ostream &operator<<(std::ostream &dest, __int128_t value) {
  std::ostream::sentry s(dest);
  if (s) {
    __uint128_t tmp = value < 0 ? -value : value;
    char buffer[128];
    char *d = std::end(buffer);
    do {
      --d;
      *d = "0123456789"[tmp % 10];
      tmp /= 10;
    } while (tmp != 0);
    if (value < 0) {
      --d;
      *d = '-';
    }
    int len = std::end(buffer) - d;
    if (dest.rdbuf()->sputn(d, len) != len) {
      dest.setstate(std::ios_base::badbit);
    }
  }
  return dest;
}
template<typename T>
vector<T> make_vec(size_t sz, T val){return std::vector<T>(sz, val);}
template<typename T, typename... Tail>
auto make_vec(size_t sz, Tail ...tail){
  return std::vector<decltype(make_vec<T>(tail...))>(sz, make_vec<T>(tail...));
}
template<typename T>
vector<T> read_vec(size_t sz){
  std::vector<T> v(sz);
  for(int i=0;i<(int)sz;i++) std::cin >> v[i];
  return v;
}
template<typename T, typename... Tail>
auto read_vec(size_t sz, Tail ...tail){
  auto v = std::vector<decltype(read_vec<T>(tail...))>(sz);
  for(int i=0;i<(int)sz;i++) v[i] = read_vec<T>(tail...);
  return v;
}
long long max(long long a, int b){return std::max(a, (long long)b);}
long long max(int a, long long b){return std::max((long long)a, b);}
long long min(long long a, int b){return std::min(a, (long long)b);}
long long min(int a, long long b){return std::min((long long)a, b);}
long long modulo(long long a, long long m){a %= m; return a < 0 ? a + m : a;}

void io_init(){
  std::cin.tie(nullptr);
  std::ios::sync_with_stdio(false);
}






template<typename Val>
struct leftist_heap{
private:
  struct node{
    node *l, *r;
    int s, sz;
    Val x;
    node(Val x): l(nullptr), r(nullptr), s(0), sz(1), x(x){}
  };
  node *root;
  static int size(node *v){
    return v ? v->sz : 0;
  }
  static bool is_null(node *v){
    return !v || !v->sz;
  }
  static node *meld_inner(node *a, node *b){
    if(is_null(a)) return b;
    if(is_null(b)) return a;
    if(a->x > b->x) std::swap(a, b);
    a->r = meld_inner(a->r, b);
    a->sz = 1 + size(a->l) + size(a->r);
    if(is_null(a->l) || a->l->s < a->r->s) std::swap(a->l, a->r);
    a->s = (is_null(a->r) ? 0 : a->r->s) + 1;
    return a;
  }
public:
  leftist_heap(): root(nullptr){}
  leftist_heap(Val x): root(new node(x)){}
  int size(){
    return (is_null(root) ? 0 : root->sz);
  }
  bool empty(){
    return size() == 0;
  }
  // マージして全要素を移動(永続でないためhの要素は全部消える)
  void meld(leftist_heap<Val> &h){
    root = meld_inner(root, h.root);
    h.root = nullptr;
  }
  Val min(){
    assert(!is_null(root));
    return root->x;
  }
  Val pop_min(){
    assert(!is_null(root));
    Val res = root->x;
    root = meld_inner(root->l, root->r);
    return res;
  }
  void push(Val x){
    root = meld_inner(root, new node(x));
  }
};


template<typename Val>
struct persistent_leftist_heap_iter;

template<typename Val>
struct persistent_leftist_heap{
private:
  struct node{
    node *l, *r;
    int s, sz;
    Val x;
    node(Val x): l(nullptr), r(nullptr), s(0), sz(1), x(x){}
  };
  static int size(node *v){
    return v ? v->sz : 0;
  }
  static bool is_null(node *v){
    return !v || !v->sz;
  }
  static node *copy_node(node *v){
    if(!v) return nullptr;
    return new node(*v);
  }
  static node *meld_inner(node *a, node *b){
    if(is_null(a)) return copy_node(b);
    if(is_null(b)) return copy_node(a);
    if(a->x > b->x) std::swap(a, b);
    a = copy_node(a);
    a->r = meld_inner(a->r, b);
    a->sz = 1 + size(a->l) + size(a->r);
    if(is_null(a->l) || a->l->s < a->r->s) std::swap(a->l, a->r);
    a->s = (is_null(a->r) ? 0 : a->r->s) + 1;
    return a;
  }
  static node *meld_build(node *a, node *b){
    if(is_null(a)) return b;
    if(is_null(b)) return a;
    if(a->x > b->x) std::swap(a, b);
    a->r = meld_inner(a->r, b);
    a->sz = 1 + size(a->l) + size(a->r);
    if(is_null(a->l) || a->l->s < a->r->s) std::swap(a->l, a->r);
    a->s = (is_null(a->r) ? 0 : a->r->s) + 1;
    return a;
  }
public:
  static node *build(const std::vector<Val> &v){
    node *res = nullptr;
    for(auto x : v) res = meld_build(res, new node(x));
    return res;
  }
  static node *make_heap(){
    return nullptr;
  }
  static node *make_heap(Val x){
    return new node(x);
  }
  static node *meld(node *a, node *b){
    return meld_inner(a, b);
  }
  static Val min(node *v){
    assert(size(v));
    return v->x;
  }
  static node *pop_min(node *v){
    assert(size(v));
    return meld(v->l, v->r);
  }
  static node *push(node *v, Val x){
    return meld(v, new node(x));
  }
  friend persistent_leftist_heap_iter<Val>;
};

template<typename Val>
struct persistent_leftist_heap_iter{
  using iter = persistent_leftist_heap_iter<Val>;
  using pheap = persistent_leftist_heap<Val>;
  using node = typename pheap::node;
  node *v;
  persistent_leftist_heap_iter(node *u): v(u){}
public:
  persistent_leftist_heap_iter(): v(nullptr){}
  persistent_leftist_heap_iter(Val x): v(pheap::make_heap(x)){}
  persistent_leftist_heap_iter(const std::vector<Val> &_v): v(pheap::build(_v)){}
  int size(){
    return pheap::size(v);
  }
  bool empty(){
    return size() == 0;
  }
  iter meld(iter &b){
    return pheap::meld(v, b.v);
  }
  Val min(){
    return pheap::min(v);
  }
  iter pop_min(){
    return pheap::pop_min(v);
  }
  iter push(Val x){
    return pheap::push(v, x);
  }
};


template<typename Idx, bool merge_adjacent = true>
struct heap_segments{
private:
  leftist_heap<std::pair<Idx, Idx>> h;
  // 左端が最小の区間をマージできる限りマージ
  void __modify(){
    assert(!h.empty());
    auto [l, r] = h.pop_min();
    while(!h.empty() && h.min().first + (!merge_adjacent) <= r){
      r = std::max(r, h.pop_min().second);
    }
    h.push({l, r});
  }
public:
  bool empty(){
    return h.empty();
  }
  // [l, r)をマージ
  // merge_adjacent: [1, 2), [2, 5)のような隣接する区間をマージするか
  void merge(Idx l, Idx r){
    h.push({l, r});
  }
  // 左端が最小の区間　
  std::pair<Idx, Idx> min(){
    __modify();
    return h.min();
  }
  // 左端が最小の区間をpopして返す
  std::pair<Idx, Idx> pop_min(){
    __modify();
    return h.pop_min();
  }
  // 任意の区間に含まれない0以上の最小要素(merge_adjacent = trueのときだけ使える)
  Idx mex(){
    static_assert(merge_adjacent);
    return empty() ? 0 : min().second;
  }
  // rの要素を全て移動(永続でないためrの要素は全て消える)
  void meld(heap_segments &r){
    h.meld(r.h);
  }
};

template<typename Idx, bool merge_adjacent = true>
struct set_segments{
  static constexpr Idx minf = std::numeric_limits<Idx>::min();
  static constexpr Idx inf = std::numeric_limits<Idx>::max();
private:
  struct node{
    int h, sz;
    Idx L, R, lensum;
    node *l, *r;
    node(Idx _L, Idx _R): h(1), sz(1), L(_L), R(_R), lensum(R - L), l(nullptr), r(nullptr){}
    int balanace_factor(){
      return (l ? l->h : 0) - (r ? r->h : 0);
    }
  };
  node *root, *tmp_node;
  int size(node *v){return v ? v->sz : 0;}
  void update(node *v){
    v->h = std::max(v->l ? v->l->h : 0,  v->r ? v->r->h : 0) + 1;
    v->sz = (v->l ? v->l->sz : 0) + (v->r ? v->r->sz : 0) + 1;
    v->lensum = (v->R - v->L) + (v->l ? v->l->lensum : 0) + (v->r ? v->r->lensum : 0);
  }
  node *rotate_right(node *v){
    node *l = v->l;
    v->l = l->r;
    l->r = v;
    update(v);
    update(l);
    return l;
  }
  node *rotate_left(node *v){
    node *r = v->r;
    v->r = r->l;
    r->l = v;
    update(v);
    update(r);
    return r;
  }
  node *balance(node *v){
    int bf = v->balanace_factor();
    assert(-2 <= bf && bf <= 2);
    if(bf == 2){
      if(v->l->balanace_factor() == -1){
        v->l = rotate_left(v->l);
        update(v);
      }
      return rotate_right(v);
    }else if(bf == -2){
      if(v->r->balanace_factor() == 1){
        v->r = rotate_right(v->r);
        update(v);
      }
      return rotate_left(v);
    }
    return v;
  }
  node *leftmost(node *v){
    while(v->l) v = v->l;
    return v;
  }
  node *rightmost(node *v){
    while(v->r) v = v->r;
    return v;
  }
  std::tuple<node*, Idx, Idx> __find(node *v, Idx k){
    Idx Lmax = minf, Rmin = inf;
    while(v){
      if(v->L <= k && k < v->R){
        return {v, v->L, v->R};
      }else if(k < v->L){
        Rmin = v->L;
        v = v->l;
      }else{
        Lmax = v->R;
        v = v->r;
      }
    }
    return {nullptr, Lmax, Rmin};
  }
  Idx __kth_point(node *v, Idx k){
    while(true){
      Idx lenl = (v->l ? v->l->lensum : 0);
      Idx lenv = lenl + (v->R - v->L);
      if(lenl <= k){
        if(k < lenv) return v->L + (k - lenl);
        k -= lenv;
        v = v->r;
      }else v = v->l;
    }
    return inf;
  }
  node *__kth_segment(node *v, int k){
    while(true){
      int szl = size(v->l);
      if(szl <= k){
        if(szl == k) return v;
        k -= szl + 1;
        v = v->r;
      }else v = v->l;
    }
  }
  int __low_count(node *v, Idx x){
    int res = 0;
    while(v){
      int szl = size(v->l);
      if(x < v->R) v = v->l;
      else v = v->r, res += szl + 1;
    }
    return res;
  }
  Idx __low_count_sum(node *v, Idx x){
    Idx res = 0;
    while(v){
      if(x <= v->L){
        v = v->l;
      }else if(v->R <= x){
        res += (v->l ? v->l->lensum : 0) + (v->R - v->L);
        v = v->r;
      }else{
        return res + (v->l ? v->l->lensum : 0) + (x - v->L);
      }
    }
    return res;
  }
   node *cut_leftmost(node *v){
    if(v->l){
      v->l = cut_leftmost(v->l);
      update(v);
      return balance(v);
    }
    tmp_node = v;
    return v->r;
  }
  node *cut_rightmost(node *v){
    if(v->r){
      v->r = cut_rightmost(v->r);
      update(v);
      return balance(v);
    }
    tmp_node = v;
    return v->l;
  }
  node *__insert(node *v, Idx l, Idx r){
    if(!v) return new node(l, r);
    if(l < v->L){
      v->l = __insert(v->l, l, r);
    }else{
      v->r = __insert(v->r, l, r);
    }
    update(v);
    return balance(v);
  }
  node *__erase(node *v, Idx l){
    if(!v) return nullptr;
    if(l < v->L){
      v->l = __erase(v->l, l);
    }else if(l > v->L){
      v->r = __erase(v->r, l);
    }else{
      if(v->r){
        v->r = cut_leftmost(v->r);
        tmp_node->l = v->l;
        tmp_node->r = v->r;
        free(v);
        update(tmp_node);
        return balance(tmp_node);
      }
      return v->l;
    }
    update(v);
    return balance(v);
  }
  void __merge(Idx l, Idx r){
    auto [L, R] = __erase_intersect(l - merge_adjacent, r + merge_adjacent);
    root = __insert(root, std::min(L, l), std::max(R, r));
  }
  // 消された中で{最小, 最大}
  std::pair<Idx, Idx> __erase_include(Idx l, Idx r){
    Idx emin = inf, emax = minf;
    for(auto [L, R] : enumerate_include(l, r)){
      root = __erase(root, L);
      emin = std::min(emin, L);
      emax = std::max(emax, R);
    }
    return {emin, emax};
  }
  // 消された中で{最小, 最大}
  std::pair<Idx, Idx> __erase_intersect(Idx l, Idx r){
    Idx emin = inf, emax = minf;
    for(auto [L, R] : enumerate_intersect(l, r)){
      root = __erase(root, L);
      emin = std::min(emin, L);
      emax = std::max(emax, R);
    }
    return {emin, emax};
  }
  void __enumerate_include(node *v, Idx l, Idx r, std::vector<std::pair<Idx, Idx>> &res){
    if(!v) return;
    if(v->l && l < v->L) __enumerate_include(v->l, l, r, res);
    if(l <= v->L && v->R <= r) res.push_back({v->L, v->R});
    if(v->r && v->R < r) __enumerate_include(v->r, l, r, res);
  }
  void __enumerate_intersect(node *v, Idx l, Idx r, std::vector<std::pair<Idx, Idx>> &res){
    if(!v) return;
    if(v->l && l < v->L) __enumerate_intersect(v->l, l, r, res);
    if(std::max(l, v->L) < std::min(r, v->R)) res.push_back({v->L, v->R});
    if(v->r && v->R < r) __enumerate_intersect(v->r, l, r, res);
  }
public:
  set_segments(): root(nullptr){}
  int size(){
    return size(root);
  }
  bool empty(){
    return size_sum(root) == 0;
  }
  // a, bが同じ区間に含まれるか
  bool same(Idx a, Idx b){
    auto [v, l, r] = find(a);
    return v && (l == std::get<1>(find(b)));
  }
  // kがいずれかの区間に含まれる場合 {1, L, R}
  // 含まれない場合 {0, L, R} (L, Rはkからいずれの区間にも含まれない座標だけを通って移動できる範囲, 何もない場合は{minf, inf})
  std::tuple<bool, Idx, Idx> find(Idx k){
    auto [v, L, R] = __find(root, k);
    return v ? std::make_tuple(true, L, R) : std::make_tuple(false, L, R);
  }
  std::pair<Idx, Idx> min(){
    assert(size());
    node *v = leftmost(root);
    return {v->L, v->R};
  }
  std::pair<Idx, Idx> max(){
    assert(size());
    node *v = rightmost(root);
    return {v->L, v->R};
  }
  // 任意の区間に含まれないかつa以上の最小要素
  Idx mex(Idx a = 0){
    static_assert(merge_adjacent);
    auto [v, L, R] = find(a);
    return v ? R : a;
  }
  // いずれかの区間に含まれるk番目(0-indexed)に小さい点. ない場合はinf
  Idx kth_point(Idx k){
    return __kth_point(root, k);
  }
  // k番目(0-indexed)に小さい区間. ない場合は{inf, inf}
  std::pair<Idx, Idx> kth_segment(int k){
    if(size() <= k) return {inf, inf};
    node *v = __kth_segment(root, k);
    return {v->L, v->R};
  }
  // [l, r)がr <= xであるような区間の数
  int low_count(Idx x){
    return __low_count(root, x);
  }
  // いずれかの区間に含まれ, かつx未満の座標の数
  Idx low_count_sum(Idx x){
    return __low_count_sum(root, x);
  }
  // [l, r)をマージ
  // merge_adjacent: [1, 2), [2, 5)のような隣接する区間をマージするか
  void merge(Idx l, Idx r){
    __merge(l, r);
  }
  // [l, r)に含まれる区間を削除
  void erase_include(Idx l, Idx r){
    __erase_include(l, r);
  }
  // [l, r)と少しでも交差する区間を削除
  void erase_intersect(Idx l, Idx r){
    __erase_intersect(l, r);
  }
  // [l, r)が完全に含む区間を列挙
  std::vector<std::pair<Idx, Idx>> enumerate_include(Idx l, Idx r){
    std::vector<std::pair<Idx, Idx>> res;
    __enumerate_include(root, l, r, res);
    return res;
  }
  // [l, r)と交差する区間を列挙
  std::vector<std::pair<Idx, Idx>> enumerate_intersect(Idx l, Idx r){
    std::vector<std::pair<Idx, Idx>> res;
    __enumerate_intersect(root, l, r, res);
    return res;
  }
  // 全区間を列挙
  std::vector<std::pair<Idx, Idx>> enumerate_all(){
    return enumerate_intersect(minf, inf);
  }
  void clear(){
    root = nullptr;
  }
  void swap(set_segments<Idx> &r){
    std::swap(root, r.root);
  }
  // rの要素を全て移動(永続でないためrの要素は全て消える)
  void meld(set_segments<Idx> &r){
    if(size() < r.size()) swap(r);
    for(auto [L, R] : r.enumerate_all()) merge(L, R);
    r.clear();
  }
};





#include <stdint.h>
static constexpr __uint128_t mask_0_64 = ((__uint128_t)1 << 64) - 1;
//static constexpr uint64_t mask_32_64 = 0xFFFFFFFF00000000;
static constexpr uint64_t mask_0_32  = 0x00000000FFFFFFFF;
static constexpr uint64_t mask_48_64 = 0xFFFF000000000000;
static constexpr uint64_t mask_32_48 = 0x0000FFFF00000000;
static constexpr uint64_t mask_16_32 = 0x00000000FFFF0000;
static constexpr uint64_t mask_0_16  = 0x000000000000FFFF;
static constexpr int TABLE_SIZE_LOG = 16, TABLE_SIZE = 1 << TABLE_SIZE_LOG;

using __table =  std::vector<std::array<int8_t, TABLE_SIZE_LOG>>;
using __table_p =  std::vector<std::array<std::pair<int8_t, int8_t>, TABLE_SIZE_LOG>>;
__table select_build(){
  __table res(TABLE_SIZE);
  for(int i = 0; i < TABLE_SIZE; i++){
    res[i].fill(-1);
    int pcnt = 0;
    for(int j = 0; j < TABLE_SIZE_LOG; j++) if((i >> j) & 1) res[i][pcnt++] = j;
  }
  return res;
}
// k-bit目以降(kも含む)に初めて現れる1の位置, 無い場合は-1
int find_next_32bit(uint32_t x, int k){
  uint32_t b = x >> k;
  if(!b) return -1;
  return k + __builtin_ctz(b);
}
// k-bit目以降(kも含む)に初めて現れる1の位置, 無い場合は-1
int find_next_64bit(uint64_t x, int k){
  uint64_t b = x >> k;
  if(!b) return -1;
  return k + __builtin_ctzll(b);
}
// 0 <= k <= 63
// k-bit目以前(kも含む)に初めて現れる1の位置, 無い場合は-1
int find_prev_64bit(uint64_t x, int k){
  uint64_t b = x << (63 - k);
  if(!b) return -1;
  return k - __builtin_clzll(b);
}
// k番目(0-indexed)の1の場所(0-indexed)を返す. 無い場合壊れる
int select_32bit(uint32_t x, int k){
  static __table table = select_build();
  int r = __builtin_popcount(x & mask_0_16);
  if(r > k) return table[x & mask_0_16][k];
  return 16 + table[(x & mask_16_32) >> 16][k - r];
}
// k番目(0-indexed)の1の場所(0-indexed)を返す. 無い場合壊れる
int select_64bit(uint64_t x, int k){
  static __table table = select_build();
  int r = __builtin_popcount(x & mask_0_32);
  if(r > k){
    int rr = __builtin_popcount(x & mask_0_16);
    if(rr > k) return table[x & mask_0_16][k];
    else return 16 + table[(x & mask_16_32) >> 16][k - rr];
  }else{
    k -= r;
    int lr = __builtin_popcountll(x & mask_32_48);
    if(lr > k) return 32 + table[(x & mask_32_48) >> 32][k];
    else return 48 + table[(x & mask_48_64) >> 48][k - lr];
  }
}
// 先頭k_bit(0 <= k <= 32)の1の数
int rank_32bit(uint32_t x, int k){
  return k == 32 ? __builtin_popcount(x) : __builtin_popcount(x & ((1ULL << k) - 1));
}
// 先頭k_bit(0 <= k <= 64)の1の数
int rank_64bit(uint64_t x, int k){
  return k == 64 ? __builtin_popcountll(x) : __builtin_popcountll(x & ((1ULL << k) - 1));
}
// 128bit
int pop_count_128bit(__uint128_t x){
  return __builtin_popcountll(x >> 64) + __builtin_popcountll(x & mask_0_64);
}
int rank_128bit(__uint128_t x, int k){
  if(k == 128) return pop_count_128bit(x);
  if(k < 64) return __builtin_popcountll((x & mask_0_64) & ((1ULL << k) - 1));
  k -= 64;
  return __builtin_popcountll(x & mask_0_64) + __builtin_popcountll((x >> 64)  & ((1ULL << k) - 1));
}
// k番目の1, 無い場合壊れる
int select1_128bit(__uint128_t x, int k){
  int left_pop = __builtin_popcountll(x & mask_0_64);
  if(left_pop > k) return select_64bit(x & mask_0_64, k);
  return 64 + select_64bit(x >> 64, k - left_pop);
}
// k番目の0, 無い場合壊れる
int select0_128bit(__uint128_t x, int k){
  __uint128_t y = ~x;
  int left_unpop = __builtin_popcountll(y & mask_0_64);
  if(left_unpop > k) return select_64bit(y & mask_0_64, k);
  return 64 + select_64bit(y >> 64, k - left_unpop);
}
// k-bit目以降(kも含む)に初めて現れる1の位置, 無い場合は-1
int find_next_128bit(__uint128_t x, int k){
  __uint128_t b = x >> k;
  if(!b) return -1;
  // 末尾64bitが0
  if(!(b & mask_0_64)) return k + 64 + __builtin_ctzll(b >> 64);
  return k + __builtin_ctzll(b);
}
// 0 <= k <= 63
// k-bit目以前(kも含む)に初めて現れる1の位置, 無い場合は-1
int find_prev_128bit(__uint128_t x, int k){
  __uint128_t b = x << (127 - k);
  if(!b) return -1;
  // 先頭64bitが0
  if(!(b >> 64)) return k - 64 - __builtin_clzll(b);
  return k - __builtin_clzll(b >> 64);
}




template<typename Val = long long>
struct binary_indexed_tree{
  int M, H;
  std::vector<Val> sum;
  binary_indexed_tree(){}
  binary_indexed_tree(int N): M(N), H(31 - __builtin_clz(M)), sum(M + 1 , 0){}
  binary_indexed_tree(const std::vector<Val> &v): M(v.size()), H(31 - __builtin_clz(M)), sum(1){
    sum.insert(sum.begin() + 1, v.begin(), v.end());
    for(int i = 1; i <= v.size(); i++){
      int nxt = i + (i & (-i));
      if(nxt <= M) sum[nxt] += sum[i];
    }
  }
  void update(int k, Val x){
    for(int i = k + 1; i <= M; i += (i & (-i))) sum[i] += x;
  }
  Val query(int r){
    Val ret = 0;
    for(int k = r; k > 0; k -= (k & (-k))) ret += sum[k];
    return ret;
  }
  Val query(int l, int r){
    return query(r) - query(l);
  }
  // sum[0, k]がx以上になるような最小のkとsum[0, k], 無い場合は{M, 総和}
  // sumが単調非減少であることが必要
  using p = std::pair<int, Val>;
  p lower_bound(Val x){
    int v = 1 << H, h = H;
    Val s = 0, t = 0;
    while(h--){
      if(M < v) v -= 1 << h;
      else if(x <= s + sum[v]) t = s + sum[v], v -= 1 << h;
      else s += sum[v], v += 1 << h;
    }
    if(v == M + 1) return {M, s};
    return (x <= s + sum[v] ? p{v - 1, s + sum[v]} : p{v, t});
  }
};


// 同じサイズのbitset同士でしか演算できない
// サイズを変更する場合resize
// n, m, im: 最大サイズ, 最大ブロック, 内部的なサイズ(これ以降に1がないことが保証されている)
// rank, selectを実行する前にブロックごとのbinary_indexed_treeを再計算
struct dynamic_bitset_rank_select{
  static constexpr int BITLEN = 64, REM = 63, SHIFT = 6;
  int n, m, im;
  std::vector<uint64_t> v;
  binary_indexed_tree<int> bit_count;

  uint64_t rightmost_mask(){
    return !(n & REM) ? ~0ULL : (1ULL << (n & REM)) - 1;
  }
  dynamic_bitset_rank_select(): n(0), m(0), im(0){}
  dynamic_bitset_rank_select(const dynamic_bitset_rank_select &r): n(r.n), m(r.m),
  im(r.im), v(r.v.begin(), r.v.begin() + r.im), bit_updated(false){}
  dynamic_bitset_rank_select(int n, bool f = 0): n(n), m((n + REM) >> SHIFT), bit_updated(false){
    if(f){
      im = m;
      v.resize(m, ~0ULL);
      v.back() &= rightmost_mask();
    }else im = 0;
  }
  // stringの左を0bit目とする
  dynamic_bitset_rank_select(const std::string &s, char one = '1'): n(s.size()), m((n + REM) >> SHIFT), im(0), bit_updated(false){
    uint64_t sum = 0;
    for(int i = 0, j = 0, t = 0; i < n; i++){
      sum += (uint64_t)(s[i] == one) << j;
      if(i == n - 1 || j == REM){
        t++;
        v.push_back(sum);
        if(sum) im = t;
        sum = j = 0;
      }else j++;
    }
  }
  int size(){
    return n;
  }
  void set(int k, bool f){
    assert(0 <= k && k < n);
    int block = k >> SHIFT;
    if(f){
      if(block >= im) im = block + 1;
      while(block >= v.size()) v.resize(std::min((int)v.size() * 2 + 1, m), 0);
      if(bit_updated && !((v[block] >> (k & REM)) & 1)) bit_count.update(block, 1);
      v[block] |= (1ULL << (k & REM));
    }else if(block < im){
      if(bit_updated && ((v[block] >> (k & REM)) & 1)) bit_count.update(block, -1);
      v[block] &= ~(1ULL << (k & REM));
    }
  }
  bool get(int k){
    return (k >> SHIFT) < im ? (v[k >> SHIFT] >> (k & REM)) & 1 : 0;
  }
  bool any(){
    if(bit_updated) return bit_count.query(m);
    for(int i = 0; i < im; i++) if(v[i]) return true;
    return false;
  }
  bool all(){
    if(bit_updated) return bit_count.query(m) == n;
    if(im != m) return false;
    for(int i = 0; i < m - 1; i++) if(v[i] != ~0ULL) return false;
    if(v[m - 1] != rightmost_mask()) return false;
    return true;
  }
  bool none(){
    return !any();
  }
  /*
  void resize(int s){
    bit_updated = false;
    m = (s + REM) >> SHIFT;
    if(m < im) v.resize(m), im = m;
    n = s;
    if(m == im) v[m - 1] &= rightmost_mask();
  }
  */
  dynamic_bitset_rank_select operator ~(){
    dynamic_bitset_rank_select res = *this;
    res.v.resize(m, ~0ULL);
    for(int i = 0; i < res.im; i++) res.v[i] = ~res.v[i];
    res.im = res.m;
    res.v[m - 1] &= res.rightmost_mask();
    return res;
  }
  dynamic_bitset_rank_select operator ^ (const dynamic_bitset_rank_select &r){
    dynamic_bitset_rank_select res(*this);
    return res ^= r;
  }
  dynamic_bitset_rank_select operator | (const dynamic_bitset_rank_select &r){
    dynamic_bitset_rank_select res(*this);
    return res |= r;
  }
  dynamic_bitset_rank_select operator & (const dynamic_bitset_rank_select &r){
    dynamic_bitset_rank_select res(*this);
    return res &= r;
  }
  dynamic_bitset_rank_select operator ^= (const dynamic_bitset_rank_select &r){
    assert(n == r.n);
    bit_updated = false;
    if(im < r.im){
      im = r.im;
      v.resize(r.im, 0);
    }
    for(int i = 0; i < r.im; i++) v[i] ^= r.v[i];
    return *this;
  }
  dynamic_bitset_rank_select operator |= (const dynamic_bitset_rank_select &r){
    assert(n == r.n);
    bit_updated = false;
    if(im < r.im){
      im = r.im;
      v.resize(r.im, 0);
    }
    for(int i = 0; i < r.im; i++) v[i] |= r.v[i];
    return *this;
  }
  dynamic_bitset_rank_select operator &= (const dynamic_bitset_rank_select &r){
    assert(n == r.n);
    bit_updated = false;
    if(im > r.im){
      im = r.im;
      v.resize(im);
    }
    for(int i = 0; i < im; i++) v[i] &= r.v[i];
    return *this;
  }
  // 全てのビットが等しく, かつサイズも同じか
  bool operator == (const dynamic_bitset_rank_select &r){
    if(n != r.n) return false;
    for(int i = 0; i < std::min(im, r.im); i++) if(v[i] != r.v[i]) return false;
    if(im > r.im){
      for(int i = r.im; i < im; i++) if(v[i]) return false;
    }else{
      for(int i = im; i < r.im; i++) if(r.v[i]) return false;
    }
    return true;
  }
  bool operator != (const dynamic_bitset_rank_select &r){
    return !(*this == r);
  }
  // l |= r << s, l = rでもok
  static void lshift_or(dynamic_bitset_rank_select &l, dynamic_bitset_rank_select &r, int s){
    assert(l.n == r.n);
    l.bit_updated = false;
    int t = s & REM, k = s >> SHIFT;
    if(!t){
      int imr = std::min(r.m, r.im + k);
      l.im = std::max(l.im, imr);
      if(imr > l.v.size()) l.v.resize(imr, 0);
      for(int i = imr - 1; i >= k; i--) l.v[i] |= r.v[i - k];
      if(imr == l.m) l.v.back() &= l.rightmost_mask();
      return;
    }
    int imr = std::min(r.m, r.im + k + 1);
    l.im = std::max(l.im, imr);
    if(imr > l.v.size()) l.v.resize(imr, 0);
    int upper_bit = 64 - t;
    for(int i = imr - 1; i >= k; i--){
      if(i == k) l.v[i] |= r.v[i - k] << t;
      else l.v[i] |= (r.v[i - k] << t) | (r.v[i - k - 1] >> upper_bit);
    }
    if(imr == l.m) l.v.back() &= l.rightmost_mask();
  }
  // l ^= r << s, l = rでもok
  static void lshift_xor(dynamic_bitset_rank_select &l, dynamic_bitset_rank_select &r, int s){
    assert(l.n == r.n);
    l.bit_updated = false;
    int t = s & REM, k = s >> SHIFT;
    if(!t){
      int imr = std::min(r.m, r.im + k);
      l.im = std::max(l.im, imr);
      if(imr > l.v.size()) l.v.resize(imr, 0);
      for(int i = imr - 1; i >= k; i--) l.v[i] ^= r.v[i - k];
      if(imr == l.m) l.v.back() &= l.rightmost_mask();
      return;
    }
    int imr = std::min(r.m, r.im + k + 1);
    l.im = std::max(l.im, imr);
    if(imr > l.v.size()) l.v.resize(imr, 0);
    int upper_bit = 64 - t;
    for(int i = imr - 1; i >= k; i--){
      if(i == k) l.v[i] ^= r.v[i - k] << t;
      else l.v[i] ^= (r.v[i - k] << t) | (r.v[i - k - 1] >> upper_bit);
    }
    if(imr == l.m) l.v.back() &= l.rightmost_mask();
  }
  // l &= r << s, l = rでもok
  static void lshift_and(dynamic_bitset_rank_select &l, dynamic_bitset_rank_select &r, int s){
    assert(l.n == r.n);
    l.bit_updated = false;
    int t = s & REM, k = s >> SHIFT;
    if(!t){
      int imr = std::min(l.im, r.im + k);
      for(int i = imr - 1; i >= 0; i--) l.v[i] &= (i >= k ? r.v[i - k] : 0);
      for(int i = l.im - 1; i >= imr; i--) l.v[i] = 0;
      l.im = imr;
      return;
    }
    int imr = std::min(l.im, r.im + k + 1);
    int upper_bit = 64 - t;
    for(int i = imr - 1; i >= 0; i--){
      if(i < k) l.v[i] = 0;
      else if(i == k) l.v[i] &= r.v[i - k] << t;
      else l.v[i] &= (r.v[i - k] << t) | (r.v[i - k - 1] >> upper_bit);
    }
    for(int i = l.im - 1; i >= imr; i--) l.v[i] = 0;
    l.im = imr;
  }
  dynamic_bitset_rank_select operator <<= (const int s){
    assert(0 <= s);
    bit_updated = false;
    int t = s & REM, k = s >> SHIFT;
    if(!t){
      im = std::min(m, im + k);
      if(im > v.size()) v.resize(im, 0);
      for(int i = im - 1; i >= 0; i--) v[i] = (i >= k ? v[i - k] : 0);
      if(im == m) v.back() &= rightmost_mask();
      return *this;
    }
    im = std::min(m, im + k + 1);
    if(im > v.size()) v.resize(im, 0);
    int upper_bit = 64 - t;
    for(int i = im - 1; i >= 0; i--){
      if(i < k) v[i] = 0;
      else if(i == k) v[i] = v[i - k] << t;
      else v[i] = (v[i - k] << t) | (v[i - k - 1] >> upper_bit);
    }
    if(im == m) v.back() &= rightmost_mask();
    return *this;
  }
  dynamic_bitset_rank_select operator >>= (const int s){
    assert(0 <= s);
    bit_updated = false;
    int t = s & REM, k = s >> SHIFT;
    im -= k;
    if(!t){
      for(int i = 0; i < im; i++) v[i] = v[i + k];
      for(int i = std::max(0, im); i < im + k; i++) v[i] = 0;
      if(im < 0) im = 0;
      return *this;
    }
    int upper_bit = 64 - t;
    for(int i = 0; i < im; i++){
      if(i + k == (int)v.size() - 1) v[i] = v[i + k] >> t;
      else v[i] = (v[i + k] >> t) | (v[i + k + 1] << upper_bit);
    }
    for(int i = std::max(0, im); i < im + k; i++) v[i] = 0;
    if(im < 0) im = 0;
    return *this;
  }
  dynamic_bitset_rank_select operator << (const int s){
    dynamic_bitset_rank_select res(*this);
    return res <<= s;
  }
  dynamic_bitset_rank_select operator >> (const int s){
    dynamic_bitset_rank_select res(*this);
    return res >>= s;
  }
  bool bit_updated;
  void bit_recalc(){
    if(bit_updated) return;
    if(bit_count.sum.size() != m + 1) bit_count = binary_indexed_tree<int>(m);
    for(int i = 0; i < m; i++){
      if(i < im) bit_count.sum[i + 1] = __builtin_popcountll(v[i]);
      else bit_count.sum[i + 1] = 0;
    }
    for(int i = 1; i <= m; i++){
      int nxt = i + (i & (-i));
      if(nxt <= m) bit_count.sum[nxt] += bit_count.sum[i];
    }
    bit_updated = true;
  }
  int pop_count(){
    if(!bit_updated) bit_recalc();
    return bit_count.query(m);
  }
  int rank1(int r){
    assert(0 <= r && r <= n);
    if(!bit_updated) bit_recalc();
    int block = r >> SHIFT;
    return bit_count.query(block) + (im <= block ? 0 : rank_64bit(v[block], r & REM));
  }
  int rank0(int r){
    assert(0 <= r && r <= n);
    return r - rank1(r);
  }
  // k個目(0-indexed)の1, 無い場合は-1
  int select1(int k){
    if(!bit_updated) bit_recalc();
    auto [b, c] = bit_count.lower_bound(k + 1);
    if(b == m) return -1;
    return (b << SHIFT) + select_64bit(v[b], k - c + __builtin_popcountll(v[b]));
  }
  // k個目(0-indexed)の0, 無い場合は-1
  int select0(int k){
    if(!bit_updated) bit_recalc();
    int x = k + 1, y = 1 << bit_count.H, h = bit_count.H;
    int s = 0, t = 0;
    while(h--){
      if(bit_count.M < y) y -= 1 << h;
      else{
        int sum_zero = s + (1 << (h + 1 + SHIFT)) - bit_count.sum[y]; // 区間長 = 2^(h + 1) * 64 = 1 << (h + 1 + SHIFT)
        if(x <= sum_zero) t = sum_zero, y -= 1 << h;
        else s = sum_zero, y += 1 << h;
      }
    }
    if(y == bit_count.M + 1) return -1;
    int b, c;
    if(x <= s + (1 << SHIFT) - bit_count.sum[y]) b = y - 1, c = s + (1 << SHIFT) - bit_count.sum[y];
    else b = y, c = t;
    int i = (b << SHIFT);
    if(b >= im) i += k - c + BITLEN;
    else i += select_64bit(~v[b], k - c + __builtin_popcountll(~v[b]));
    return i >= n ? -1 : i;
  }
  // i以降(i含む)の1, 無い場合は-1
  int find_next1(int i){
    assert(0 <= i && i < n);
    if((i >> SHIFT) >= im) return -1;
    int _i = i & REM, _j = find_next_64bit(v[i >> SHIFT], _i);
    if(_j != -1) return i - _i + _j;
    return select1(rank1(i));
  }
  // i以前(i含む)の1, 無い場合は-1
  int find_prev1(int i){
    assert(0 <= i && i < n);
    if((i >> SHIFT) >= im){
      if(!im) return -1;
      i = (im << SHIFT) - 1;
    }
    int _i = i & REM, _j = find_prev_64bit(v[i >> SHIFT], _i);
    if(_j != -1) return i - _i + _j;
    int r = rank1(i + 1);
    if(!r) return -1;
    return select1(r - 1);
  }
  // i以降(i含む)の0, 無い場合は-1
  int find_next0(int i){
    assert(0 <= i && i < n);
    if((i >> SHIFT) >= im) return i;
    int _i = i & REM, _j = find_next_64bit(~v[i >> SHIFT], _i);
    if(_j != -1){
      int res = i - _i + _j;
      return res >= n ? -1 : res;
    }
    return select0(rank0(i));
  }
  // i以前(i含む)の0, 無い場合は-1
  int find_prev0(int i){
    assert(0 <= i && i < n);
    if((i >> SHIFT) >= im) return i;
    int _i = i & REM, _j = find_prev_64bit(~v[i >> SHIFT], _i);
    if(_j != -1) return i - _i + _j;
    int r = rank0(i + 1);
    if(!r) return -1;
    return select0(r - 1);
  }
};
std::ostream &operator<<(std::ostream &dest, const dynamic_bitset_rank_select &v){
  if(!v.n) return dest;
  int cnt = 0;
  for(int i = 0; i < v.m; i++){
    if(i >= v.im){
      for(int j = 0; j < 64 && cnt < v.n; j++, cnt++) dest << 0;
    }else{
      for(int j = 0; j < 64 && cnt < v.n; j++, cnt++) dest << ((v.v[i] >> j) & 1);
    }
    if(i != v.m - 1) dest << ' ';
  }
  return dest;
}


int main(){
  io_init();
  int h, w, n;
  std::cin >> h >> w >> n;
  auto v = make_vec<int>(h, w, 0);
  range(i, 0, n){
    int a, b;
    std::cin >> a >> b;
    a--, b--;
    v[a][b] = 1;
  }

  ll ans = 0;
  vector<int> d(w, 0);
  range(i, 0, h){
    range(j, 0, w) d[j] = (v[i][j] ? 0 : d[j] + 1);
    vector<pair<int, int>> E;
    range(j, 0, w) if(i >= d[j]) E.push_back({i - d[j], j});
    sort(allof(E));
    reverse(allof(E));
    
    
    int S = w * (w + 1) / 2;
    /*
    set_segments<int> seg;
    seg.merge(0, w);
    for(int j = i, idx = 0; j >= 0; j--){
      while(idx < E.size() && E[idx].first == j){
        auto [_, col] = E[idx++];
        auto [f, l, r] = seg.find(col);
        //seg.erase_intersect(l, r);
        //if(l < col) seg.merge(l, col);
        //if(col + 1 < r) seg.merge(col + 1, r);
        S -= (r - l) * (r - l + 1) / 2;
        S += (col - l) * (col - l + 1) / 2;
        S += (r - col - 1) * (r - col) / 2;
      }
      ans += S;
    }
    */
    dynamic_bitset_rank_select bit(w, 1);
    for(int j = i, idx = 0; j >= 0; j--){
      while(idx < E.size() && E[idx].first == j){
        auto [_, col] = E[idx++];
        int l = bit.find_prev0(col) + 1;
        int r = bit.find_next0(col);
        if(r == -1) r = w;
        bit.set(col, 0);
        S -= (r - l) * (r - l + 1) / 2;
        S += (col - l) * (col - l + 1) / 2;
        S += (r - col - 1) * (r - col) / 2;
      }
      ans += S;
    }
  }
  std::cout << ans << '\n';
}