//#define _GLIBCXX_DEBUG

//#pragma GCC target("avx2")
//#pragma GCC optimize("O3")
//#pragma GCC optimize("unroll-loops")

#include<bits/stdc++.h>
using namespace std;


#ifdef LOCAL
#include <debug_print.hpp>
#define OUT(...) debug_print::multi_print(#__VA_ARGS__, __VA_ARGS__)
#else
#define OUT(...) (static_cast<void>(0))
#endif

#define endl '\n'
#define lfs cout<<fixed<<setprecision(15)
#define ALL(a)  (a).begin(),(a).end()
#define ALLR(a)  (a).rbegin(),(a).rend()
#define UNIQUE(a) (a).erase(unique((a).begin(),(a).end()),(a).end())
#define spa << " " <<
#define fi first
#define se second
#define MP make_pair
#define MT make_tuple
#define PB push_back
#define EB emplace_back
#define rep(i,n,m) for(ll i = (n); i < (ll)(m); i++)
#define rrep(i,n,m) for(ll i = (ll)(m) - 1; i >= (ll)(n); i--)

namespace template_tute{
  using ll = long long;
  using ld = long double;
  const ll MOD1 = 1e9+7;
  const ll MOD9 = 998244353;
  const ll INF = 1e18;
  using P = pair<ll, ll>;
  template<typename T> using PQ = priority_queue<T>;
  template<typename T> using QP = priority_queue<T,vector<T>,greater<T>>;
  template<typename T1, typename T2>bool chmin(T1 &a,T2 b){if(a>b){a=b;return true;}else return false;}
  template<typename T1, typename T2>bool chmax(T1 &a,T2 b){if(a<b){a=b;return true;}else return false;}
  ll median(ll a,ll b, ll c){return a+b+c-max<ll>({a,b,c})-min<ll>({a,b,c});}
  void ans1(bool x){if(x) cout<<"Yes"<<endl;else cout<<"No"<<endl;}
  void ans2(bool x){if(x) cout<<"YES"<<endl;else cout<<"NO"<<endl;}
  void ans3(bool x){if(x) cout<<"Yay!"<<endl;else cout<<":("<<endl;}
  template<typename T1,typename T2>void ans(bool x,T1 y,T2 z){if(x)cout<<y<<endl;else cout<<z<<endl;}  
  template<typename T1,typename T2,typename T3>void anss(T1 x,T2 y,T3 z){ans(x!=y,x,z);};  
  template<typename T>void debug(const T &v,ll h,ll w,string sv=" "){for(ll i=0;i<h;i++){cout<<v[i][0];for(ll j=1;j<w;j++)cout<<sv<<v[i][j];cout<<endl;}};
  template<typename T>void debug(const T &v,ll n,string sv=" "){if(n!=0)cout<<v[0];for(ll i=1;i<n;i++)cout<<sv<<v[i];cout<<endl;};
  template<typename T>void debug(const vector<T>&v){debug(v,v.size());}
  template<typename T>void debug(const vector<vector<T>>&v){for(auto &vv:v)debug(vv,vv.size());}
  template<typename T>void debug(stack<T> st){while(!st.empty()){cout<<st.top()<<" ";st.pop();}cout<<endl;}
  template<typename T>void debug(queue<T> st){while(!st.empty()){cout<<st.front()<<" ";st.pop();}cout<<endl;}
  template<typename T>void debug(deque<T> st){while(!st.empty()){cout<<st.front()<<" ";st.pop_front();}cout<<endl;}
  template<typename T>void debug(PQ<T> st){while(!st.empty()){cout<<st.top()<<" ";st.pop();}cout<<endl;}
  template<typename T>void debug(QP<T> st){while(!st.empty()){cout<<st.top()<<" ";st.pop();}cout<<endl;}
  template<typename T>void debug(const set<T>&v){for(auto z:v)cout<<z<<" ";cout<<endl;}
  template<typename T>void debug(const multiset<T>&v){for(auto z:v)cout<<z<<" ";cout<<endl;}
  template<typename T,size_t size>void debug(const array<T, size> &a){for(auto z:a)cout<<z<<" ";cout<<endl;}
  template<typename T,typename V>void debug(const map<T,V>&v){for(auto z:v)cout<<"["<<z.first<<"]="<<z.second<<",";cout<<endl;}
  template<typename T>vector<vector<T>>vec(ll x, ll y, T w){vector<vector<T>>v(x,vector<T>(y,w));return v;}
  vector<ll>dx={1,-1,0,0,1,1,-1,-1};vector<ll>dy={0,0,1,-1,1,-1,1,-1};
  template<typename T>vector<T> make_v(size_t a,T b){return vector<T>(a,b);}
  template<typename... Ts>auto make_v(size_t a,Ts... ts){return vector<decltype(make_v(ts...))>(a,make_v(ts...));}
  template<typename T1, typename T2>ostream &operator<<(ostream &os, const pair<T1, T2>&p){return os << "(" << p.first << "," << p.second << ")";}
  template<typename T>ostream &operator<<(ostream &os, const vector<T> &v){os<<"[";for(auto &z:v)os << z << ",";os<<"]"; return os;}
  template<typename T>void rearrange(vector<int>&ord, vector<T>&v){
    auto tmp = v;
    for(int i=0;i<tmp.size();i++)v[i] = tmp[ord[i]];
  }
  template<typename Head, typename... Tail>void rearrange(vector<int>&ord,Head&& head, Tail&&... tail){
    rearrange(ord, head);
    rearrange(ord, tail...);
  }
  template<typename T> vector<int> ascend(const vector<T>&v){
    vector<int>ord(v.size());iota(ord.begin(),ord.end(),0);
    sort(ord.begin(),ord.end(),[&](int i,int j){return make_pair(v[i],i)<make_pair(v[j],j);});
    return ord;
  }
  template<typename T> vector<int> descend(const vector<T>&v){
    vector<int>ord(v.size());iota(ord.begin(),ord.end(),0);
    sort(ord.begin(),ord.end(),[&](int i,int j){return make_pair(v[i],-i)>make_pair(v[j],-j);});
    return ord;
  }
  template<typename T> vector<T> inv_perm(const vector<T>&ord){
    vector<T>inv(ord.size());
    for(int i=0;i<ord.size();i++)inv[ord[i]] = i;
    return inv;
  }
  ll FLOOR(ll n,ll div){assert(div>0);return n>=0?n/div:(n-div+1)/div;}
  ll CEIL(ll n,ll div){assert(div>0);return n>=0?(n+div-1)/div:n/div;}
  ll digitsum(ll n){ll ret=0;while(n){ret+=n%10;n/=10;}return ret;}
  ll modulo(ll n,ll d){return (n%d+d)%d;};
  template<typename T>T min(const vector<T>&v){return *min_element(v.begin(),v.end());}
  template<typename T>T max(const vector<T>&v){return *max_element(v.begin(),v.end());}
  template<typename T>T acc(const vector<T>&v){return accumulate(v.begin(),v.end(),T(0));};
  template<typename T>T reverse(const T &v){return T(v.rbegin(),v.rend());};
  //mt19937 mt(chrono::steady_clock::now().time_since_epoch().count());
  int popcount(ll x){return __builtin_popcountll(x);};
  int poplow(ll x){return __builtin_ctzll(x);};
  int pophigh(ll x){return 63 - __builtin_clzll(x);};
  template<typename T>T poll(queue<T> &q){auto ret=q.front();q.pop();return ret;};
  template<typename T>T poll(priority_queue<T> &q){auto ret=q.top();q.pop();return ret;};
  template<typename T>T poll(QP<T> &q){auto ret=q.top();q.pop();return ret;};
  template<typename T>T poll(stack<T> &s){auto ret=s.top();s.pop();return ret;};
  ll MULT(ll x,ll y){if(LLONG_MAX/x<=y)return LLONG_MAX;return x*y;}
  ll POW2(ll x, ll k){ll ret=1,mul=x;while(k){if(mul==LLONG_MAX)return LLONG_MAX;if(k&1)ret=MULT(ret,mul);mul=MULT(mul,mul);k>>=1;}return ret;}
  ll POW(ll x, ll k){ll ret=1;for(int i=0;i<k;i++){if(LLONG_MAX/x<=ret)return LLONG_MAX;ret*=x;}return ret;}
  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;
  }
  namespace converter{
    int dict[500];
    const string lower="abcdefghijklmnopqrstuvwxyz";
    const string upper="ABCDEFGHIJKLMNOPQRSTUVWXYZ";
    const string digit="0123456789";
    const string digit1="123456789";
    void regi_str(const string &t){
      for(int i=0;i<t.size();i++){
        dict[t[i]]=i;
      }
    }
    void regi_int(const string &t){
      for(int i=0;i<t.size();i++){
        dict[i]=t[i];
      }
    }
    vector<int>to_int(const string &s,const string &t){
      regi_str(t);
      vector<int>ret(s.size());
      for(int i=0;i<s.size();i++){
        ret[i]=dict[s[i]];
      }
      return ret;
    }
    vector<int>to_int(const string &s){
      auto t=s;
      sort(t.begin(),t.end());
      t.erase(unique(t.begin(),t.end()),t.end());
      return to_int(s,t);
    }
    
    vector<vector<int>>to_int(const vector<string>&s,const string &t){
      regi_str(t);
      vector<vector<int>>ret(s.size(),vector<int>(s[0].size()));
      for(int i=0;i<s.size();i++){
        for(int j=0;j<s[0].size();j++){
          ret[i][j]=dict[s[i][j]];
        }
      }
      return ret;
    }
    vector<vector<int>>to_int(const vector<string>&s){
      string t;
      for(int i=0;i<s.size();i++){
        t+=s[i];
      }
      sort(t.begin(),t.end());t.erase(unique(t.begin(),t.end()),t.end());
      return to_int(s,t);
    }
    string to_str(const vector<int>&s,const string &t){
      regi_int(t);
      string ret;
      for(auto z:s)ret+=dict[z];
      return ret;
    }
    vector<string> to_str(const vector<vector<int>>&s,const string &t){
      regi_int(t);
      vector<string>ret(s.size());
      for(int i=0;i<s.size();i++){
        for(auto z:s[i])ret[i]+=dict[z];
      }
      return ret;
    }
  }
  template< typename T = int >
  struct edge {
    int to;
    T cost;
    int id;
    edge():to(-1),id(-1){};
    edge(int to, T cost = 1, int id = -1):to(to), cost(cost), id(id){}
    operator int() const { return to; }
  };

  template<typename T>
  using Graph = vector<vector<edge<T>>>;
  template<typename T>
  Graph<T>revgraph(const Graph<T> &g){
    Graph<T>ret(g.size());
    for(int i=0;i<g.size();i++){
      for(auto e:g[i]){
        int to = e.to;
        e.to = i;
        ret[to].push_back(e);
      }
    }
    return ret;
  }
  template<typename T>
  Graph<T> readGraph(int n,int m,int indexed=1,bool directed=false,bool weighted=false){
    Graph<T> ret(n);
    for(int es = 0; es < m; es++){
      int u,v;
      T w=1;
      cin>>u>>v;u-=indexed,v-=indexed;
      if(weighted)cin>>w;
      ret[u].emplace_back(v,w,es);
      if(!directed)ret[v].emplace_back(u,w,es);
    }
    return ret;
  }
  template<typename T>
  Graph<T> readParent(int n,int indexed=1,bool directed=true){
    Graph<T>ret(n);
    for(int i=1;i<n;i++){
      int p;cin>>p;
      p-=indexed;
      ret[p].emplace_back(i);
      if(!directed)ret[i].emplace_back(p);
    }
    return ret;
  }
}
using namespace template_tute;
namespace GeneralSlopeTrick{
  using I = ll;
  const I minf = -1.5e9; //無限小の座標
  const I max_identity = minf * 2; //座標の単位元
  // minf での値はオーバーフローしても問題ない(はず)

  //base: https://nyaannyaan.github.io/library/rbst/lazy-reversible-rbst.hpp
  template <typename Node>
  struct RBSTBase {
    //using Ptr = Node *;
    // template <typename... Args>
    // inline Ptr my_new(Args... args) {
    //   return new Node(args...);
    // }
    // inline void my_del(Ptr t) { delete t; }
    // inline Ptr make_tree() const { return nullptr; }

    // for avoiding memory leak, activate below
    
    using Ptr = shared_ptr<Node>;
    template <typename... Args>
    inline Ptr my_new(Args... args) {
      return make_shared<Node>(args...);
    }
    inline void my_del(Ptr t) {}
    Ptr make_tree() {return Ptr();}
    

    int size(Ptr t) const { return count(t); }

    using Key = decltype(Node::key);
    //checkを満たす境界位置 0～size
    template<typename check>
    int find_first(Ptr t, check &C) {
      int ret = 0;
      Key now;
      if (!C(sum(t)))return count(t);
      while(1){
        if(!t)return ret;
        push(t);
        if(t->l)push(t->l);
        if(C(f(now, sum(t->l)))){
          t = t->l;
        }
        else{
          now = f(now, sum(t->l));
          ret += count(t->l);
          now = f(now, t->key);
          if(C(now))return ret;
          ret++;
          t = t->r;
        }
      }
    }
    Ptr merge(Ptr l, Ptr r) {
      if (!l || !r) return l ? l : r;
      if (int((rng() * (l->cnt + r->cnt)) >> 32) < l->cnt) {
        push(l);
        l->r = merge(l->r, r);
        return update(l);
      } else {
        push(r);
        r->l = merge(l, r->l);
        return update(r);
      }
    }

    pair<Ptr, Ptr> split(Ptr t, int k) {
      if (!t) return {nullptr, nullptr};
      push(t);
      if (k <= count(t->l)) {
        auto s = split(t->l, k);
        t->l = s.second;
        return {s.first, update(t)};
      } else {
        auto s = split(t->r, k - count(t->l) - 1);
        t->r = s.first;
        return {update(t), s.second};
      }
    }

    Ptr build(int l, int r, const vector<decltype(Node::key)> &v) {
      if (l + 1 == r) return my_new(v[l]);
      int m = (l + r) >> 1;
      Ptr pm = my_new(v[m]);
      if (l < m) pm->l = build(l, m, v);
      if (m + 1 < r) pm->r = build(m + 1, r, v);
      return update(pm);
    }

    Ptr build(const vector<decltype(Node::key)> &v) {
      return build(0, (int)v.size(), v);
    }

    template <typename... Args>
    void insert(Ptr &t, int k, const Args &... args) {
      auto x = split(t, k);
      t = merge(merge(x.first, my_new(args...)), x.second);
    }

    template <typename... Args>
    void push_back(Ptr &t, const Args &... args) {
      t = merge(t, my_new(args...));
    }

    template <typename... Args>
    void push_front(Ptr &t, const Args &... args) {
      t = merge(my_new(args...), t);
    }

    void erase(Ptr &t, int k) {
      auto x = split(t, k);
      auto y = split(x.second, 1);
      my_del(y.first);
      t = merge(x.first, y.second);
    }
    inline int count(const Ptr t) const { return t ? t->cnt : 0; }
    inline Key sum(const Ptr t) const { return t ? t->sum : Key(); }


  //protected:
    static uint64_t rng() {
      static uint64_t x_ = 88172645463325252ULL;
      return x_ ^= x_ << 7, x_ ^= x_ >> 9, x_ & 0xFFFFFFFFull;
    }


    virtual void push(Ptr) = 0;

    virtual Ptr update(Ptr) = 0;
  };

  template <typename T, typename E>
  struct LazyRBSTNode {
    typename RBSTBase<LazyRBSTNode>::Ptr l, r;
    T key, sum;
    E lazy;
    int cnt;

    LazyRBSTNode(const T &t = T(), const E &e = E())
        : l(), r(), key(t), sum(t), lazy(e), cnt(1){}
  };

  template <typename T, typename E, T (*f)(T, T), T (*g)(T, E), E (*h)(E, E)>
  struct LazyRBST : RBSTBase<LazyRBSTNode<T, E>> {
    using Node = LazyRBSTNode<T, E>;
    using base = RBSTBase<LazyRBSTNode<T, E>>;
    using base::merge;
    using base::split;
    using typename base::Ptr;

    LazyRBST() = default;

    T fold(Ptr &t, int a, int b) {
      auto x = split(t, a);
      auto y = split(x.second, b - a);
      auto ret = sum(y.first);
      t = merge(x.first, merge(y.first, y.second));
      return ret;
    }

    void apply(Ptr &t, int a, int b, const E &e) {
      auto x = split(t, a);
      auto y = split(x.second, b - a);
      propagate(y.first, e);
      t = merge(x.first, merge(y.first, y.second));
    }
    inline T sum(const Ptr t) const { return t ? t->sum : T(); }
  //protected:

    Ptr update(Ptr t) override {
      push(t);
      t->cnt = 1;
      t->sum = t->key;
      if (t->l) t->cnt += t->l->cnt, t->sum = f(t->l->sum, t->sum);
      if (t->r) t->cnt += t->r->cnt, t->sum = f(t->sum, t->r->sum);
      return t;
    }

    void push(Ptr t) override {
      if (t->lazy != E()) {
        if (t->l) propagate(t->l, t->lazy);
        if (t->r) propagate(t->r, t->lazy);
        t->lazy = E();
      }
    }

    void propagate(Ptr t, const E &x) {
      t->lazy = h(t->lazy, x);
      t->key = g(t->key, x);
      t->sum = g(t->sum, x);
    }
  };

  struct T{
    I pos_mx, grad, sum;
    T():pos_mx(max_identity), grad(0), sum(0){}
    T(I pos_mx, I grad, I sum): pos_mx(pos_mx), grad(grad), sum(sum){}
  };
  using E = I;
  T f(T a,T b){
    return T(max(a.pos_mx, b.pos_mx), a.grad + b.grad, a.sum + b.sum);
  }
  T g(T a,E b){
    return T(a.pos_mx + b, a.grad, a.sum + a.grad * b);
  }
  E h(E a,E b){
    return a + b;
  }
  struct SlopeTrick{
    LazyRBST<T, E, f, g, h>bbst;
    using Ptr = LazyRBST<T, E, f, g, h>::Ptr;
    Ptr root;
    I minf_val, minf_grad;
    SlopeTrick(){
      root = bbst.build(vector<T>({T(0, 0, 0)}));
      minf_val = 0;
      minf_grad = 0;
    }
    void insert(I pos,I val){
      auto check = [&](T s){
        return s.pos_mx > pos;
      };
      int idx = bbst.find_first(root, check);
      bbst.insert(root, idx, T(pos, val, pos * val));
    }
    void add_all(I a){
      minf_val += a;
    }
    // add c(x-a)_+   _____/
    void add_xma(I a, I c = 1){
      insert(a, c);
    }
    // add c(a-x)_+   \_____
    void add_amx(I a, I c = 1) {
      minf_grad -= c;
      minf_val += c * (a - minf);
      insert(a, c);
    }
    // add |x-a|     \____/
    void add_abs(I a, I c = 1) { add_xma(a, c), add_amx(a, c); }
    void add_abs_fast(I a, I c = 1) {
      minf_grad -= c;
      minf_val += c * (a - minf);
      insert(a, 2*c);
    }
    pair<Ptr, Ptr>zero_split(){
      auto check = [&](T s){
        return s.grad + minf_grad >= 0;
      };
      int idx = bbst.find_first(root, check);
      auto [l, r] = bbst.split(root, idx + 1);
      return make_pair(l, r);
    }
    tuple<Ptr, Ptr, Ptr>zero_split3(){
      auto check = [&](T s){
        return s.grad + minf_grad >= 0;
      };
      int idx = bbst.find_first(root, check);
      auto [l, r] = bbst.split(root, idx);
      auto [rl, rr] = bbst.split(r, 1);
      return make_tuple(l, rl, rr);
    }
    //  chmin right side \_/ -> \__
    void chmin_right() {
      auto [l, m, r] = zero_split3();
      auto lsum = bbst.sum(l);
      m->key.grad = -minf_grad - lsum.grad;
      m->key.sum = (-minf_grad - lsum.grad) * m->key.pos_mx;
      m = bbst.update(m);
      root = bbst.merge(l, m);
    }
    //  chmin left side  \_/ -> __/
    void chmin_left() {
      auto [l, m, r] = zero_split3();
      auto lsum = bbst.sum(l);
      minf_val += (minf_grad + lsum.grad) * m->key.pos_mx - (minf_grad * minf + lsum.sum);
      m->key.grad += minf_grad + lsum.grad;
      m->key.sum = m->key.grad * m->key.pos_mx;
      minf_grad = 0;
      m = bbst.update(m);
      root = bbst.merge(m, r);
    }
    // move left with cost c
    // limit が -1 なら無限、そうでなければ limit だけ左に移動できる
    // 最小値の位置が無限遠になるときはすべて move系/add系で処理する(他は壊れがち)
    void move_left(I c, I limit = -1){
      assert(limit >= -1);
      if(minf_grad >= -c)return;
      auto root_grad = bbst.sum(root).grad;
      if(minf_grad + root_grad < -c){
        assert(limit != -1); // 無限に小さくなる
        bbst.propagate(root, -limit);
        minf_val += (minf_grad + c) * limit;
        return;
      }
      auto check = [&](T s){
        return s.grad >= -c - minf_grad;
      };
      int idx = bbst.find_first(root, check);
      auto [l, r] = bbst.split(root, idx);
      auto [rl, rr] = bbst.split(r, 1);
      auto lsum=bbst.sum(l);
      I dec = -c - (lsum.grad + minf_grad);
      rl->key.grad -= dec;
      rl->key.sum = rl->key.grad * rl->key.pos_mx;
      rl = bbst.update(rl);
      if(limit == -1){
        minf_grad = -c;
        minf_val -= lsum.sum - minf * lsum.grad;
        minf_val -= (rl->key.pos_mx - minf) * dec;
      }
      else{
        minf_val -= (-c - minf_grad) * limit;
        bbst.push_back(l, T(rl->key.pos_mx, dec, dec * rl->key.pos_mx));
        bbst.propagate(l, -limit);
        rl = bbst.merge(l, rl);
      }
      root = bbst.merge(rl, rr);
    }
    // move right with cost c
    // limit が -1 なら無限、そうでなければ limit だけ右に移動できる
    void move_right(I c, I limit = -1){
      assert(limit >= -1);
      if(minf_grad + root->sum.grad <= c)return;
      if(minf_grad > c){
        assert(limit != -1); // 無限に小さくなる
        bbst.propagate(root, limit);
        minf_val -= (minf_grad - c) * limit;
        return;
      }
      auto check = [&](T s){
        return s.grad + minf_grad >= c;
      };
      int idx = bbst.find_first(root, check);
      auto [l, r] = bbst.split(root, idx);
      auto [rl, rr] = bbst.split(r, 1);
      auto rrsum=bbst.sum(rr);
      I dec = (minf_grad + bbst.sum(l).grad + rl->key.grad) - c;
      rl->key.grad -= dec;
      rl->key.sum = rl->key.grad * rl->key.pos_mx;
      rl = bbst.update(rl);
      if(limit != -1){
        bbst.push_front(rr, T(rl->key.pos_mx, dec, dec * rl->key.pos_mx));
        bbst.propagate(rr, limit);
        rl = bbst.merge(rl, rr);
      }
      root = bbst.merge(l, rl);
    }
    void shift(I k){
      bbst.propagate(root, k);
      minf_val -= k * minf_grad;
    }
    I get_min(){
      auto [l, r] = zero_split();
      auto lsum = bbst.sum(l);
      I ret = minf_val + (minf_grad + lsum.grad) * lsum.pos_mx - (minf_grad * minf + lsum.sum);
      root = bbst.merge(l, r);
      return ret;
    }
    I calc(I pos){
      auto check = [&](T s){
        return s.pos_mx > pos;
      };
      int idx = bbst.find_first(root, check);
      auto [l, r] = bbst.split(root, idx);
      auto lsum = bbst.sum(l);
      I ret = minf_val + (minf_grad + lsum.grad) * pos - (minf_grad * minf + lsum.sum);
      root = bbst.merge(l, r);
      return ret;
    }
    void slide_min(I a,I b){
      assert(a <= b);
      shift(a);
      auto [l, m, r] = zero_split3();
      auto lsum = bbst.sum(l);
      I nxt_grad = m->key.grad - (-minf_grad - lsum.grad);
      m->key.grad -= nxt_grad;
      m->key.sum = m->key.grad * m->key.pos_mx;
      m = bbst.update(m);
      bbst.insert(m, 1, T(m->key.pos_mx + b - a, nxt_grad, (m->key.pos_mx + b - a) * nxt_grad));
      if(r)bbst.propagate(r, b - a);
      root = bbst.merge(m, r);root = bbst.merge(l, root);
    }
    void print_grad(){
      for(int i=0;i<bbst.count(root);i++){
        auto s = bbst.fold(root, i, i+1);
        cout << "[" << s.pos_mx << "]=" << s.grad << endl;
      }
    }
  void print(I l,I r){
      for(I i=l;i<=r;i++){
        cout<<calc(i)<<" ";
      }
      cout<<endl;
    }
  };
}
void solve(){
	ll res=0,buf=0;
  bool judge = true;

  ll m,n;cin>>m>>n;
  vector<ll>a(m);
  rep(i,0,m)cin>>a[i];
  vector<ll>b(n);
  rep(i,0,n)cin>>b[i];
  vector<P>p;
  rep(i,0,m)p.EB(a[i],0);
  rep(i,0,n)p.EB(b[i],1);
  sort(ALL(p));
  rep(k,1,m+1){
    GeneralSlopeTrick::SlopeTrick st;
    st.add_abs_fast(0,1e18);
    ll cnt=0;
    rep(i,0,p.size()){
      if(p[i].se==0){
        cnt++;
      }
      else{
        st.move_right(0,k);
      }
      if(i+1<p.size())st.add_abs_fast(cnt,p[i+1].fi-p[i].fi);
      //if(k==1)st.print(0,4);
    }
    cout<<st.calc(m)<<endl;
  }
}

int main(){
  cin.tie(nullptr);
  ios_base::sync_with_stdio(false);
  ll res=0,buf=0;
  bool judge = true;
  int T = 1;
  //cin>>T;
  while(T--){
    solve();
  }
  return 0;
}