#include<bits/stdc++.h>
#include <immintrin.h>
#pragma GCC target("avx2")
#pragma GCC optimize("O3")
#pragma GCC optimize("unroll-loops")
#define FOR1(a) for (ll _ = 0; _ < ll(a); ++_)
#define FOR2(i, a) for (ll i = 0; i < ll(a); ++i)
#define FOR3(i, a, b) for (ll i = a; i < ll(b); ++i)
#define FOR4(i, a, b, c) for (ll i = a; i < ll(b); i += (c))
#define FOR1_R(a) for (ll i = (a)-1; i >= ll(0); --i)
#define FOR2_R(i, a) for (ll i = (a)-1; i >= ll(0); --i)
#define FOR3_R(i, a, b) for (ll i = (b)-1; i >= ll(a); --i)
#define overload4(a, b, c, d, e, ...) e
#define overload3(a, b, c, d, ...) d
#define FOR(...) overload4(__VA_ARGS__, FOR4, FOR3, FOR2, FOR1)(__VA_ARGS__)
#define FOR_R(...) overload3(__VA_ARGS__, FOR3_R, FOR2_R, FOR1_R)(__VA_ARGS__)
#define sz(c) ((int)(c).size())
#define ten(x) ((int)1e##x)
#define all(v) (v).begin(), (v).end()
using namespace std;
using ll=long long;
using i128=__int128;
using P = pair<ll,ll>;
const long double PI=acos(-1);
const ll INF=1e18;
const int inf=1e9;
struct Edge {
    ll to;
    ll cost;
};
using Graph=vector<vector<Edge>>;
template <typename T>
bool chmax(T &a,const T& b){
  if (a<b){
    a=b;
    return true;
  }
  return false;
}
template <typename T>
bool chmin(T &a,const T& b){
  if (a>b){
    a=b;
    return true;
  }
  return false;
}
template<int MOD> struct Fp{
  ll val;
  constexpr Fp(long long v = 0) noexcept : val(v % MOD) {
    if (val < 0) val += MOD;
  }
  static constexpr int getmod() { return MOD; }
  constexpr Fp operator - () const noexcept {
    return val ? MOD - val : 0;
  }
  constexpr Fp operator + (const Fp& r) const noexcept { return Fp(*this) += r; }
  constexpr Fp operator - (const Fp& r) const noexcept { return Fp(*this) -= r; }
  constexpr Fp operator * (const Fp& r) const noexcept { return Fp(*this) *= r; }
  constexpr Fp operator / (const Fp& r) const noexcept { return Fp(*this) /= r; }
  constexpr Fp& operator += (const Fp& r) noexcept {
    val += r.val;
    if (val >= MOD) val -= MOD;
    return *this;
  }
  constexpr Fp& operator -= (const Fp& r) noexcept {
    val -= r.val;
    if (val < 0) val += MOD;
    return *this;
  }
  constexpr Fp& operator *= (const Fp& r) noexcept {
    val = val * r.val % MOD;
    return *this;
  }
  constexpr Fp& operator /= (const Fp& r) noexcept {
    ll a = r.val, b = MOD, u = 1, v = 0;
    while (b) {
      ll t = a / b;
      a -= t * b, swap(a, b);
      u -= t * v, swap(u, v);
    }
    val = val * u % MOD;
    if (val < 0) val += MOD;
    return *this;
  }
  constexpr bool operator == (const Fp& r) const noexcept {
    return this->val == r.val;
  }
  constexpr bool operator != (const Fp& r) const noexcept {
    return this->val != r.val;
  }
  constexpr bool operator < (const Fp& r) const noexcept {
    return this->val < r.val;
  }
  friend constexpr istream& operator >> (istream& is, Fp<MOD>& x) noexcept {
    is >> x.val;
    x.val %= MOD;
    if (x.val < 0) x.val += MOD;
    return is;
  }
  friend constexpr ostream& operator << (ostream& os, const Fp<MOD>& x) noexcept {
    return os << x.val;
  }
  friend constexpr Fp<MOD> modpow(const Fp<MOD>& a, long long n) noexcept {
    Fp<MOD> res=1,r=a;
    while(n){
      if(n&1) res*=r;
      r*=r;
      n>>=1;
    }
    return res;
  }
  friend constexpr Fp<MOD> modinv(const Fp<MOD>& r) noexcept {
        long long a = r.val, b = MOD, u = 1, v = 0;
        while (b) {
            long long t = a / b;
            a -= t * b, swap(a, b);
            u -= t * v, swap(u, v);
        }
        return Fp<MOD>(u);
  }
  ll get(){
      return val;
  }
  explicit operator bool()const{
		return val;
  }
};
template< uint32_t mod, bool fast = false >
struct MontgomeryModInt {
  using mint = MontgomeryModInt;
  using i32 = int32_t;
  using i64 = int64_t;
  using u32 = uint32_t;
  using u64 = uint64_t;
 
  static constexpr u32 get_r() {
    u32 ret = mod;
    for(i32 i = 0; i < 4; i++) ret *= 2 - mod * ret;
    return ret;
  }
 
  static constexpr u32 r = get_r();
  static constexpr u32 n2 = -u64(mod) % mod;
 
  static_assert(r * mod == 1, "invalid, r * mod != 1");
  static_assert(mod < (1 << 30), "invalid, mod >= 2 ^ 30");
  static_assert((mod & 1) == 1, "invalid, mod % 2 == 0");
 
  u32 a;
 
  MontgomeryModInt() : a{} {}
 
  MontgomeryModInt(const i64 &x)
      : a(reduce(u64(fast ? x : (x % mod + mod)) * n2)) {}
 
  static constexpr u32 reduce(const u64 &b) {
    return u32(b >> 32) + mod - u32((u64(u32(b) * r) * mod) >> 32);
  }
 
  constexpr mint& operator+=(const mint &p) {
    if(i32(a += p.a - 2 * mod) < 0) a += 2 * mod;
    return *this;
  }
 
  constexpr mint& operator-=(const mint &p) {
    if(i32(a -= p.a) < 0) a += 2 * mod;
    return *this;
  }
 
  constexpr mint& operator*=(const mint &p) {
    a = reduce(u64(a) * p.a);
    return *this;
  }
 
  constexpr mint& operator/=(const mint &p) {
    *this *= modinv(p);
    return *this;
  }
 
  constexpr mint operator-() const { return mint() - *this; }
 
  constexpr mint operator+(const mint &p) const { return mint(*this) += p; }
 
  constexpr mint operator-(const mint &p) const { return mint(*this) -= p; }
 
  constexpr mint operator*(const mint &p) const { return mint(*this) *= p; }
 
  constexpr mint operator/(const mint &p) const { return mint(*this) /= p; }
 
  constexpr bool operator==(const mint &p) const { return (a >= mod ? a - mod : a) == (p.a >= mod ? p.a - mod : p.a); }
 
  constexpr bool operator!=(const mint &p) const { return (a >= mod ? a - mod : a) != (p.a >= mod ? p.a - mod : p.a); }
 
  u32 get() const {
    u32 ret = reduce(a);
    return ret >= mod ? ret - mod : ret;
  }
 
  friend constexpr MontgomeryModInt<mod> modpow(const MontgomeryModInt<mod> &x,u64 n) noexcept {
    MontgomeryModInt<mod> ret(1), mul(x);
    while(n > 0) {
      if(n & 1) ret *= mul;
      mul *= mul;
      n >>= 1;
    }
    return ret;
  }
 
  friend constexpr MontgomeryModInt<mod> modinv(const MontgomeryModInt<mod> &r) noexcept {
        u64 a = r.get(), b = mod, u = 1, v = 0;
        while (b) {
            long long t = a / b;
            a -= t * b, swap(a, b);
            u -= t * v, swap(u, v);
        }
        return MontgomeryModInt<mod>(u);
  }
 
  friend ostream &operator<<(ostream &os, const mint &p) {
    return os << p.get();
  }
 
  friend istream &operator>>(istream &is, mint &a) {
    i64 t;
    is >> t;
    a = mint(t);
    return is;
  }
  static constexpr u32 getmod() { return mod; }
};
ll mod(ll a, ll mod) {
    return (a%mod+mod)%mod;
}
ll modpow(ll a,ll n,ll mod){
  ll res=1;
  a%=mod;
  while (n>0){
    if (n & 1) res*=a;
    a *= a;
    a%=mod;
    n >>= 1;
    res%=mod;
  }
  return res;
}
vector<P> prime_factorize(ll N) {
  vector<P> res;
  for (ll a = 2; a * a <= N; ++a) {
    if (N % a != 0) continue;
    ll ex = 0;
    while(N % a == 0){
      ++ex;
      N /= a;
    }
    res.push_back({a, ex});
  }
  if (N != 1) res.push_back({N, 1});
  return res;
}
ll modinv(ll a, ll mod) {
    ll b = mod, u = 1, v = 0;
    while (b) {
        ll t = a/b;
        a -= t * b, swap(a, b);
        u -= t * v, swap(u, v);
    }
    u %= mod;
    if (u < 0) u += mod;
    return u;
}
ll extGcd(ll a, ll b, ll &p, ll &q) {  
    if (b == 0) { p = 1; q = 0; return a; }  
    ll d = extGcd(b, a%b, q, p);  
    q -= a/b * p;  
    return d;  
}
P ChineseRem(const vector<ll> &b, const vector<ll> &m) {
  ll r = 0, M = 1;
  for (int i = 0; i < (int)b.size(); ++i) {
    ll p, q;
    ll d = extGcd(M, m[i], p, q);
    if ((b[i] - r) % d != 0) return make_pair(0, -1);
    ll tmp = (b[i] - r) / d * p % (m[i]/d);
    r += M * tmp;
    M *= m[i]/d;
  }
  return make_pair(mod(r, M), M);
}
#include <unistd.h>

namespace fastio {
#define FASTIO
struct has_write_impl {
  template <class T>
  static auto check(T &&x) -> decltype(x.write(), std::true_type{});
 
  template <class T>
  static auto check(...) -> std::false_type;
};
 
template <class T>
class has_write : public decltype(has_write_impl::check<T>(std::declval<T>())) {
};
 
struct has_read_impl {
  template <class T>
  static auto check(T &&x) -> decltype(x.read(), std::true_type{});
 
  template <class T>
  static auto check(...) -> std::false_type;
};
 
template <class T>
class has_read : public decltype(has_read_impl::check<T>(std::declval<T>())) {};
 
struct Scanner {
  FILE *fp;
  char line[(1 << 15) + 1];
  size_t st = 0, ed = 0;
  void reread() {
    memmove(line, line + st, ed - st);
    ed -= st;
    st = 0;
    ed += fread(line + ed, 1, (1 << 15) - ed, fp);
    line[ed] = '\0';
  }
  bool succ() {
    while (true) {
      if (st == ed) {
        reread();
        if (st == ed) return false;
      }
      while (st != ed && isspace(line[st])) st++;
      if (st != ed) break;
    }
    if (ed - st <= 50) {
      bool sep = false;
      for (size_t i = st; i < ed; i++) {
        if (isspace(line[i])) {
          sep = true;
          break;
        }
      }
      if (!sep) reread();
    }
    return true;
  }
  template <class T, enable_if_t<is_same<T, string>::value, int> = 0>
  bool read_single(T &ref) {
    if (!succ()) return false;
    while (true) {
      size_t sz = 0;
      while (st + sz < ed && !isspace(line[st + sz])) sz++;
      ref.append(line + st, sz);
      st += sz;
      if (!sz || st != ed) break;
      reread();
    }
    return true;
  }
  template <class T, enable_if_t<is_integral<T>::value, int> = 0>
  bool read_single(T &ref) {
    if (!succ()) return false;
    bool neg = false;
    if (line[st] == '-') {
      neg = true;
      st++;
    }
    ref = T(0);
    while (isdigit(line[st])) { ref = 10 * ref + (line[st++] & 0xf); }
    if (neg) ref = -ref;
    return true;
  }
  template <typename T,
            typename enable_if<has_read<T>::value>::type * = nullptr>
  inline bool read_single(T &x) {
    x.read();
    return true;
  }
  bool read_single(double &ref) {
    string s;
    if (!read_single(s)) return false;
    ref = std::stod(s);
    return true;
  }
  bool read_single(char &ref) {
    string s;
    if (!read_single(s) || s.size() != 1) return false;
    ref = s[0];
    return true;
  }
  template <class T>
  bool read_single(vector<T> &ref) {
    for (auto &d: ref) {
      if (!read_single(d)) return false;
    }
    return true;
  }
  template <class T, class U>
  bool read_single(pair<T, U> &p) {
    return (read_single(p.first) && read_single(p.second));
  }
  template <size_t N = 0, typename T>
  void read_single_tuple(T &t) {
    if constexpr (N < std::tuple_size<T>::value) {
      auto &x = std::get<N>(t);
      read_single(x);
      read_single_tuple<N + 1>(t);
    }
  }
  template <class... T>
  bool read_single(tuple<T...> &tpl) {
    read_single_tuple(tpl);
    return true;
  }
  void read() {}
  template <class H, class... T>
  void read(H &h, T &... t) {
    bool f = read_single(h);
    assert(f);
    read(t...);
  }
  Scanner(FILE *fp) : fp(fp) {}
};
 
struct Printer {
  Printer(FILE *_fp) : fp(_fp) {}
  ~Printer() { flush(); }
 
  static constexpr size_t SIZE = 1 << 15;
  FILE *fp;
  char line[SIZE], small[50];
  size_t pos = 0;
  void flush() {
    fwrite(line, 1, pos, fp);
    pos = 0;
  }
  void write(const char val) {
    if (pos == SIZE) flush();
    line[pos++] = val;
  }
  template <class T, enable_if_t<is_integral<T>::value, int> = 0>
  void write(T val) {
    if (pos > (1 << 15) - 50) flush();
    if (val == 0) {
      write('0');
      return;
    }
    if (val < 0) {
      write('-');
      val = -val; // todo min
    }
    size_t len = 0;
    while (val) {
      small[len++] = char(0x30 | (val % 10));
      val /= 10;
    }
    for (size_t i = 0; i < len; i++) { line[pos + i] = small[len - 1 - i]; }
    pos += len;
  }
  void write(const string s) {
    for (char c: s) write(c);
  }
  void write(const char *s) {
    size_t len = strlen(s);
    for (size_t i = 0; i < len; i++) write(s[i]);
  }
  void write(const double x) {
    ostringstream oss;
    oss << fixed << setprecision(15) << x;
    string s = oss.str();
    write(s);
  }
  void write(const long double x) {
    ostringstream oss;
    oss << fixed << setprecision(15) << x;
    string s = oss.str();
    write(s);
  }
  template <typename T,
            typename enable_if<has_write<T>::value>::type * = nullptr>
  inline void write(T x) {
    x.write();
  }
  template <class T>
  void write(const vector<T> val) {
    auto n = val.size();
    for (size_t i = 0; i < n; i++) {
      if (i) write(' ');
      write(val[i]);
    }
  }
  template <class T, class U>
  void write(const pair<T, U> val) {
    write(val.first);
    write(' ');
    write(val.second);
  }
  template <size_t N = 0, typename T>
  void write_tuple(const T t) {
    if constexpr (N < std::tuple_size<T>::value) {
      if constexpr (N > 0) { write(' '); }
      const auto x = std::get<N>(t);
      write(x);
      write_tuple<N + 1>(t);
    }
  }
  template <class... T>
  bool write(tuple<T...> tpl) {
    write_tuple(tpl);
    return true;
  }
  template <class T, size_t S>
  void write(const array<T, S> val) {
    auto n = val.size();
    for (size_t i = 0; i < n; i++) {
      if (i) write(' ');
      write(val[i]);
    }
  }
  void write(i128 val) {
    string s;
    bool negative = 0;
    if (val < 0) {
      negative = 1;
      val = -val;
    }
    while (val) {
      s += '0' + int(val % 10);
      val /= 10;
    }
    if (negative) s += "-";
    reverse(all(s));
    if (sz(s) == 0) s = "0";
    write(s);
  }
};
Scanner scanner = Scanner(stdin);
Printer printer = Printer(stdout);
void flush() { printer.flush(); }
void print() { printer.write('\n'); }
template <class Head, class... Tail>
void print(Head &&head, Tail &&... tail) {
  printer.write(head);
  if (sizeof...(Tail)) printer.write(' ');
  print(forward<Tail>(tail)...);
}
void print2() {}
template <class Head, class... Tail>
void print2(Head &&head, Tail &&... tail){
    printer.write(head);
    if(sizeof...(Tail)) printer.write(' ');
    print2(forward<Tail>(tail)...);
}
void read() {}
template <class Head, class... Tail>
void read(Head &head, Tail &... tail) {
  scanner.read(head);
  read(tail...);
}
} // namespace fastio
using fastio::print;
using fastio::flush;
using fastio::read;
using fastio::printer;
using fastio::print2;
constexpr bool is_prime(int n) {
    if (n <= 1) return false;
    if (n == 2 || n == 7 || n == 61) return true;
    if (n % 2 == 0) return false;
    long long d = n - 1;
    while (d % 2 == 0) d /= 2;
    constexpr long long bases[3] = {2, 7, 61};
    for (long long a : bases) {
        long long t = d;
        long long y = modpow(a, t, n);
        while (t != n - 1 && y != 1 && y != n - 1) {
            y = y * y % n;
            t <<= 1;
        }
        if (y != n - 1 && t % 2 == 0) {
            return false;
        }
    }
    return true;
}
template< typename T >
struct Combination {
  vector< T > _fact, _rfact, _inv;
 
  Combination(int sz) : _fact(sz + 1), _rfact(sz + 1), _inv(sz + 1) {
    _fact[0] = _rfact[sz] = _inv[0] = 1;
    for(int i = 1; i <= sz; i++) _fact[i] = _fact[i - 1] * i;
    _rfact[sz] /= _fact[sz];
    for(int i = sz - 1; i >= 0; i--) _rfact[i] = _rfact[i + 1] * (i + 1);
    for(int i = 1; i <= sz; i++) _inv[i] = _rfact[i] * _fact[i - 1];
  }
 
  inline T fact(int k) const { return _fact[k]; }
 
  inline T rfact(int k) const { return _rfact[k]; }
 
  inline T inv(int k) const { return _inv[k]; }
 
  T P(int n, int r) const {
    if(r < 0 || n < r) return 0;
    return fact(n) * rfact(n - r);
  }
 
  T C(int p, int q) const {
    if(q < 0 || p < q) return 0;
    return fact(p) * rfact(q) * rfact(p - q);
  }
 
  T H(int n, int r) const {
    if(n < 0 || r < 0) return (0);
    return r == 0 ? 1 : C(n + r - 1, r);
  }
};
ll rand_int(ll l, ll r) { //[l, r]
	#ifdef LOCAL
	static mt19937_64 gen;
	#else
    static random_device rd;
    static mt19937_64 gen(rd());
    #endif
    return uniform_int_distribution<ll>(l, r)(gen);
}
template<class T,T (*op)(T,T),T (*e)()> struct SegmentTree{
  int n;
  vector<T> dat;
  SegmentTree(int N){
    n=1;
    while(n<N)n*=2;
    dat.assign(2*n,e());
  }
  void add(int k,T x){
    k+=n;
    dat[k]+=x;
    while(k){
      k>>=1;
      dat[k]=op(dat[k*2],dat[k*2+1]);
    }
  }
  void apply(int k,T x){
    k+=n;
    dat[k]=op(dat[k],x);
    while(k){
      k>>=1;
      dat[k]=op(dat[k*2],dat[k*2+1]);
    }
  }
  void set(int k,T x){
    k+=n;
    dat[k]=x;
    while(k){
      k>>=1;
      dat[k]=op(dat[k*2],dat[k*2+1]);
    }
  }
  T query(int l,int r){
    T prodl=e(),prodr=e();
    l+=n;
    r+=n;
    while(l<r){
      if(l&1) prodl=op(prodl,dat[l++]);
      if(r&1) prodr=op(dat[--r],prodr);
      l>>=1;
      r>>=1;
    }
    return op(prodl,prodr);
  }
};
template<class S,S (*op)(S,S),S (*e)(),class F,S (*mapping)(F,S),F (*composition)(F,F),F (*id)()>
struct LazySegTree{
  private:
  int _n,size=1,idx=0;
  vector<S>seq;
  vector<F>lazy;
  void update(int k){seq[k]=op(seq[2*k],seq[2*k+1]);}
  void all_apply(int k,F f){
    seq[k]=mapping(f,seq[k]);
    if(k<size)lazy[k]=composition(f,lazy[k]);
  }
  void eval(int k){
    all_apply(2*k,lazy[k]);
    all_apply(2*k+1,lazy[k]);
    lazy[k]=id();
  }
  public:
  LazySegTree(int n):LazySegTree(vector<S>(n,e())){}
  LazySegTree(const vector<S>&v):_n(int(v.size())){
    while(size<_n)size<<=1,idx++;
    seq=vector<S>(2*size,e());
    lazy=vector<F>(2*size,id());
    for(int i=0;i<_n;i++)seq[size+i]=v[i];
    for(int i=size-1;i>=1;i--)update(i);
  }
  void set(int p,S x){
    p+=size;
    for(int i=idx;i>=1;i--)eval(p>>i);
    seq[p]=x;
    for(int i=1;i<=idx;i++)update(p>>i);
  }
  void add(int p,S x){
    p+=size;
    for(int i=idx;i>=1;i--)eval(p>>i);
    seq[p]+=x;
    for(int i=1;i<=idx;i++)update(p>>i);
  }
  S operator[](int p){
    p+=size;
    for(int i=idx;i>=1;i--)eval(p>>i);
    return seq[p];
  }
  S query(int l,int r){
    if(l==r)return e();
    S sml=e(),smr=e();
    l+=size,r+=size;
    for(int i=idx;i>=1;i--){
      if(((l>>i)<<i)!=l)eval(l>>i);
      if(((r>>i)<<i)!=r)eval(r>>i);
    }
    while(l<r){
      if(l&1)sml=op(sml,seq[l++]);
      if(r&1)smr=op(seq[--r],smr);
      l>>=1,r>>=1;
    }
    return op(sml,smr);
  }
  S all_query()const{return seq[1];}
  void apply(int p,F f){
    p+=size;
    for(int i=idx;i>=1;i--)eval(p>>i);
    seq[p]=mapping(f,seq[p]);
    for(int i=1;i<=idx;i++)update(p>>i);
  }
  void apply(int l,int r,F f){
    if(l==r)return ;
    l+=size;
    r+=size;
    for(int i=idx;i>=1;i--){
      if(((l>>i)<<i)!=l)eval(l>>i);
      if(((r>>i)<<i)!=r)eval((r-1)>>i);
    }
    int l2=l,r2=r;
    while(l<r){
      if(l&1)all_apply(l++,f);
      if(r&1)all_apply(--r,f);
      l>>=1;
      r>>=1;
    }
    l=l2,r=r2;
    for(int i=1;i<=idx;i++){
      if(((l>>i)<<i)!=l)update(l>>i);
      if(((r>>i)<<i)!=r)update((r-1)>>i);
    }
  }
};
template< typename flow_t >
struct MFGraph{
  static_assert(is_integral< flow_t >::value, "template parameter flow_t must be integral type");
  const flow_t INF;
  struct edge {
    int to;
    flow_t cap;
    int rev;
    bool isrev;
    int idx;
  };
 
  vector< vector< edge > > graph;
  vector< int > min_cost, iter;
  flow_t max_cap;
  int SZ;
 
  explicit MFGraph(int V) : INF(numeric_limits< flow_t >::max()), graph(V), max_cap(0), SZ(V) {}
 
  void add_edge(int from, int to, flow_t cap, int idx = -1) {
    max_cap = max(max_cap, cap);
    graph[from].emplace_back((edge) {to, cap, (int) graph[to].size(), false, idx});
    graph[to].emplace_back((edge) {from, 0, (int) graph[from].size() - 1, true, idx});
  }
 
  bool build_augment_path(int s, int t, const flow_t &base) {
    min_cost.assign(graph.size(), -1);
    queue< int > que;
    min_cost[s] = 0;
    que.push(s);
    while(!que.empty() && min_cost[t] == -1) {
      int p = que.front();
      que.pop();
      for(auto &e : graph[p]) {
        if(e.cap >= base && min_cost[e.to] == -1) {
          min_cost[e.to] = min_cost[p] + 1;
          que.push(e.to);
        }
      }
    }
    return min_cost[t] != -1;
  }
 
  flow_t find_augment_path(int idx, const int t, flow_t base, flow_t flow) {
    if(idx == t) return flow;
    flow_t sum = 0;
    for(int &i = iter[idx]; i < (int)graph[idx].size(); i++) {
      edge &e = graph[idx][i];
      if(e.cap >= base && min_cost[idx] < min_cost[e.to]) {
        flow_t d = find_augment_path(e.to, t, base, min(flow - sum, e.cap));
        if(d > 0) {
          e.cap -= d;
          graph[e.to][e.rev].cap += d;
          sum += d;
          if(flow - sum < base) break;
        }
      }
    }
    return sum;
  }
 
  flow_t max_flow(int s, int t) {
    if(max_cap == flow_t(0)) return flow_t(0);
    flow_t flow = 0;
    for(int i = 63 - __builtin_clzll(max_cap); i >= 0; i--) {
      flow_t now = flow_t(1) << i;
      while(build_augment_path(s, t, now)) {
        iter.assign(graph.size(), 0);
        flow += find_augment_path(s, t, now, INF);
      }
    }
    return flow;
  }
 
  //after max_flow(s,t)
  vector<bool> min_cut(int s) {
        vector<bool> visited(SZ);
        queue<int> que;
        que.push(s);
        while (!que.empty()) {
            int p = que.front();
            que.pop();
            visited[p] = true;
            for (auto e : graph[p]) {
                if (e.cap && !visited[e.to]) {
                    visited[e.to] = true;
                    que.push(e.to);
                }
            }
        }
        return visited;
    }
  
  void output() {
    for(int i = 0; i < graph.size(); i++) {
      for(auto &e : graph[i]) {
        if(e.isrev) continue;
        auto &rev_e = graph[e.to][e.rev];
        cout << i << "->" << e.to << " (flow: " << rev_e.cap << "/" << e.cap + rev_e.cap << ")" << endl;
      }
    }
  }
};
template< typename flow_t, typename cost_t >
struct MCFGraph{
  struct edge {
    int to;
    flow_t cap;
    cost_t cost;
    int rev;
    bool isrev;
  };
 
  vector< vector< edge > > graph;
  vector< cost_t > potential, min_cost;
  vector< int > prevv, preve;
  const cost_t INF;
 
  MCFGraph(int V) : graph(V), INF(numeric_limits< cost_t >::max()) {}
 
  void add_edge(int from, int to, flow_t cap, cost_t cost) {
    graph[from].emplace_back((edge) {to, cap, cost, (int) graph[to].size(), false});
    graph[to].emplace_back((edge) {from, 0, -cost, (int) graph[from].size() - 1, true});
  }
 
  cost_t min_cost_flow(int s, int t, flow_t f) {
    int V = (int) graph.size();
    cost_t ret = 0;
    using Pi = pair< cost_t, int >;
    priority_queue< Pi, vector< Pi >, greater< Pi > > que;
    potential.assign(V, 0);
    preve.assign(V, -1);
    prevv.assign(V, -1);
 
    while(f > 0) {
      min_cost.assign(V, INF);
      que.emplace(0, s);
      min_cost[s] = 0;
      while(!que.empty()) {
        Pi p = que.top();
        que.pop();
        if(min_cost[p.second] < p.first) continue;
        for(int i = 0; i < (int)graph[p.second].size(); i++) {
          edge &e = graph[p.second][i];
          cost_t nextCost = min_cost[p.second] + e.cost + potential[p.second] - potential[e.to];
          if(e.cap > 0 && min_cost[e.to] > nextCost) {
            min_cost[e.to] = nextCost;
            prevv[e.to] = p.second, preve[e.to] = i;
            que.emplace(min_cost[e.to], e.to);
          }
        }
      }
      if(min_cost[t] == INF) return -1;
      for(int v = 0; v < V; v++) potential[v] += min_cost[v];
      flow_t addflow = f;
      for(int v = t; v != s; v = prevv[v]) {
        addflow = min(addflow, graph[prevv[v]][preve[v]].cap);
      }
      f -= addflow;
      ret += addflow * potential[t];
      for(int v = t; v != s; v = prevv[v]) {
        edge &e = graph[prevv[v]][preve[v]];
        e.cap -= addflow;
        graph[v][e.rev].cap += addflow;
      }
    }
    return ret;
  }
 
  void output() {
    for(int i = 0; i < graph.size(); i++) {
      for(auto &e : graph[i]) {
        if(e.isrev) continue;
        auto &rev_e = graph[e.to][e.rev];
        cout << i << "->" << e.to << " (flow: " << rev_e.cap << "/" << rev_e.cap + e.cap << ")" << endl;
      }
    }
  }
};
struct UnionFind{
  int n;
  vector<int> data;
  vector<int> edge_num;
  UnionFind(int N) : n(N) , data(N,-1) , edge_num(N,0){}
  int root(int x){ // データxが属する木の根を再帰で得る：root(x) = {xの木の根}
    return data[x]<0?x:data[x]=root(data[x]);
  }
  void unite(int x, int y) {
    x=root(x);
    y=root(y);
    if(x!=y){
      if (data[x]>data[y]) swap(x, y);
      data[x]+=data[y];
      data[y]=x;
    }
    if(x!=y){
      edge_num[x]+=edge_num[y];
      edge_num[y]=0;
    }
    edge_num[x]+=1;
  }
  bool same(int x, int y){ // 2つのデータx, yが属する木が同じならtrueを返す
    return root(x)==root(y);
  }
  int size(int x){
    return -data[root(x)];
  }
  int edge(int x){
    return edge_num[root(x)];
  }
  vector<int> get_roots(){
    vector<int> res;
    for(int i=0;i<n;i++){
      if(data[i]<0) res.push_back(i);
    }
    return res;
  }
};
vector<ll> enum_divisors(ll N){
  vector<ll> res;
  for (ll i = 1; i * i <= N; ++i){
    if (N % i == 0){
       res.push_back(i);
       // 重複しないならば i の相方である N/i も push
       if (N/i != i) res.push_back(N/i);
    }
  }
  // 小さい順に並び替える
  sort(res.begin(), res.end());
  return res;
}
using mint=MontgomeryModInt<998244353>;
#define in(...) read(__VA_ARGS__)
#define LL(...) ll __VA_ARGS__;in(__VA_ARGS__)
#define INT(...) int __VA_ARGS__;in(__VA_ARGS__)
#define STR(...) string __VA_ARGS__;in(__VA_ARGS__)
#define outln(...) print(__VA_ARGS__)
#define out(...) print2(__VA_ARGS__)
#define outspace(...) print2(__VA_ARGS__),printer.write(' ')
#define rall(v) (v).rbegin(), (v).rend()
#define fi first
#define se second
#define endl '\n'
using i128=__int128_t;
struct LowerHull{
    struct Point{ ll x; i128 y; };
    vector<Point> h;
    void add(ll x,i128 y){
        Point p{x,y};
        if(!h.empty()&&h.back().x==x) return;
        while (h.size() >= 2) {
            auto a=h[h.size()-2],b=h.back();
            if((i128)(b.x-a.x)*(y-a.y)>(b.y-a.y)*(x-a.x)) break;
            h.pop_back();
        }
        h.push_back(p);
    }
    //min(y + k*x)
    i128 query(i128 k) const {
        auto f=[&](int i){
            return h[i].y+k*h[i].x;
        };
        int l=0,r=(int)h.size()-1;
        while(l<r){
            int m=(l+r)/2;
            if(f(m)<=f(m+1)) r=m;
            else l=m+1;
        }
        return f(l);
    }
};
void slv(){
    LL(n,m,q);
    vector<ll> a(n),xs;
    FOR(i,n){
        in(a[i]);
        a[i]%=m;
        xs.push_back(a[i]);
    }
    vector<pair<int,ll>> qs(q);
    FOR(i,q){
        in(qs[i].fi,qs[i].se);
        qs[i].fi--;
        qs[i].se%=m;
        xs.push_back(qs[i].se);
    }
    sort(all(xs));
    xs.erase(unique(all(xs)),xs.end());
    auto id=[&](ll x){
        return int(lower_bound(all(xs),x)-xs.begin());
    };
    int K=xs.size(),B=max(1,(int)(2*sqrt(K)));
    int nb=(K+B-1)/B;
    vector<int> c(K),lazyC(nb);
    vector<ll> p(K),lazyP(nb);
    vector<LowerHull> hull(nb);
    FOR(i,n) c[id(a[i])]++;
    ll S=0,D=0;
    int cnt=0;
    FOR(i,K){
        int f=c[i];
        D+=(ll)((i128)f*((i128)cnt*xs[i]-S));
        S+=(ll)f*xs[i];
        cnt+=f;
        c[i]=cnt;
        p[i]=S;
    }
    auto rebuild=[&](int b){
        hull[b].h.clear();
        FOR(i,b*B,min(K,(b+1)*B)) hull[b].add(c[i],(i128)2*n*p[i]-(i128)m*c[i]*c[i]);
    };
    auto suffix=[&](int i,int dc,ll dp){
        int b=i/B;
        if(i%B){
            FOR(j,i,min(K,(b+1)*B)){
                c[j]+=dc;
                p[j]+=dp;
            }
            rebuild(b++);
        }
        for(;b<nb;b++){
            lazyC[b]+=dc;
            lazyP[b]+=dp;
        }
    };
    FOR(b,nb) rebuild(b);
    for(auto [i,x] : qs){
        ll old=a[i];
        if(old!=x){
            int u=id(old),v=id(x);
            ll cu=c[u]+lazyC[u/B],pu=p[u]+lazyP[u/B];
            ll cv=c[v]+lazyC[v/B],pv=p[v]+lazyP[v/B];
            int t=(old<=x);
            i128 dec=(i128)old*(2*cu-n)+S-2*pu;
            i128 inc=(i128)x*(2*(cv-t)-(n-1))+S-old-2*(pv-t*old);
            D+=(ll)(inc-dec);
            suffix(u,-1,-old);
            suffix(v,+1,x);
            S+=x-old;
            a[i]=x;
        }
        i128 k=(i128)n*m-2*S;
        i128 best=(i128)1<<120;
        FOR(b,nb){
            ll dc=lazyC[b],dp=lazyP[b];
            chmin(best,hull[b].query(k-(i128)2*m*dc)+(i128)2*n*dp-(i128)m*dc*dc+k*dc);
        }
        outln(D+(ll)best);
    }
}
int main(){
    ios::sync_with_stdio(false);
    cin.tie(nullptr);
    int t=1;
    //in(t);
    while(t--){
        slv();
    }
}