Problem: Vova decided to clean his room. The room can be represented as the coordinate axis OX. There are n piles of trash in the room, coordinate of the i-th pile is the integer p_i. All piles have different coordinates.

Let's define a total cleanup as the following process. The goal of this process is to collect all the piles in no more than two different x coordinates. To achieve this goal, Vova can do several (possibly, zero) moves. During one move, he can choose some x and move all piles from x to x+1 or x-1 using his broom. Note that he can't choose how many piles he will move.

Also, there are two types of queries:

  * 0 x — remove a pile of trash from the coordinate x. It is guaranteed that there is a pile in the coordinate x at this moment. 
  * 1 x — add a pile of trash to the coordinate x. It is guaranteed that there is no pile in the coordinate x at this moment. 



Note that it is possible that there are zero piles of trash in the room at some moment.

Vova wants to know the minimum number of moves he can spend if he wants to do a total cleanup before any queries. He also wants to know this number of moves after applying each query. Queries are applied in the given order. Note that the total cleanup doesn't actually happen and doesn't change the state of piles. It is only used to calculate the number of moves.

For better understanding, please read the Notes section below to see an explanation for the first example.

Input

The first line of the input contains two integers n and q (1 ≤ n, q ≤ 10^5) — the number of piles in the room before all queries and the number of queries, respectively.

The second line of the input contains n distinct integers p_1, p_2, ..., p_n (1 ≤ p_i ≤ 10^9), where p_i is the coordinate of the i-th pile.

The next q lines describe queries. The i-th query is described with two integers t_i and x_i (0 ≤ t_i ≤ 1; 1 ≤ x_i ≤ 10^9), where t_i is 0 if you need to remove a pile from the coordinate x_i and is 1 if you need to add a pile to the coordinate x_i. It is guaranteed that for t_i = 0 there is such pile in the current set of piles and for t_i = 1 there is no such pile in the current set of piles.

Output

Print q+1 integers: the minimum number of moves Vova needs to do a total cleanup before the first query and after each of q queries.

Examples

Input


5 6
1 2 6 8 10
1 4
1 9
0 6
0 10
1 100
1 50


Output


5
7
7
5
4
8
49


Input


5 8
5 1 2 4 3
0 1
0 2
0 3
0 4
0 5
1 1000000000
1 1
1 500000000


Output


3
2
1
0
0
0
0
0
499999999

Note

Consider the first example.

Initially, the set of piles is [1, 2, 6, 8, 10]. The answer before the first query is 5 because you can move all piles from 1 to 2 with one move, all piles from 10 to 8 with 2 moves and all piles from 6 to 8 with 2 moves.

After the first query, the set becomes [1, 2, 4, 6, 8, 10]. Then the answer is 7 because you can move all piles from 6 to 4 with 2 moves, all piles from 4 to 2 with 2 moves, all piles from 2 to 1 with 1 move and all piles from 10 to 8 with 2 moves.

After the second query, the set of piles becomes [1, 2, 4, 6, 8, 9, 10] and the answer is the same (and the previous sequence of moves can be applied to the current set of piles).

After the third query, the set of piles becomes [1, 2, 4, 8, 9, 10] and the answer is 5 because you can move all piles from 1 to 2 with 1 move, all piles from 2 to 4 with 2 moves, all piles from 10 to 9 with 1 move and all piles from 9 to 8 with 1 move.

After the fourth query, the set becomes [1, 2, 4, 8, 9] and the answer is almost the same (the previous sequence of moves can be applied without moving piles from 10).

After the fifth query, the set becomes [1, 2, 4, 8, 9, 100]. You can move all piles from 1 and further to 9 and keep 100 at its place. So the answer is 8.

After the sixth query, the set becomes [1, 2, 4, 8, 9, 50, 100]. The answer is 49 and can be obtained with almost the same sequence of moves as after the previous query. The only difference is that you need to move all piles from 50 to 9 too.

Solution (in JAVA):

Solution:
/*
    Java might be good for Competitive Programming but Python still rocks!!!!!!!!!


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                         ppppp  ppppppppppppppppppp
                      ppppppp    ppppppppppppppppppppp
                      pppppppp  pppppppppppppppppppppp
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    ppppppppppppppppppppppppppppppppppppppppppppppp    pppppppppppppppppppppppp
   pppppppppppppppppppppppppppppppppppppppppppppp     pppppppppppppppppppppppppp
  ppppppppppppppppppppppppppppppppppppppppppppp      pppppppppppppppppppppppppppp
  pppppppppppppppppppppppppppppppp               pppppppppppppppppppppppppppppppp
  pppppppppppppppppppppppppppp     pppppppppppppppppppppppppppppppppppppppppppppp
  ppppppppppppppppppppppppppp    pppppppppppppppppppppppppppppppppppppppppppppppp
    pppppppppppppppppppppppp  pppppppppppppppppppppppppppppppppppppppppppppppppp
     ppppppppppppppppppppppp  ppppppppppppppppppppppppppppppppppppppppppppppppp
      pppppppppppppppppppppp  ppppppppppppppppppppppppppppppppppppppppppppppp
       ppppppppppppppppppppp  ppppppppppppppppppppppppppppppppppppppppppppp
                              pppppppppppppppppppppppp
                              pppppppppppppppppppppppppppppppp
                              pppppppppppppppppppppp  pppppppp
                              ppppppppppppppppppppp    ppppppp
                                 ppppppppppppppppppp  ppppp
                                    pppppppppppppppppppp
*/


import java.util.*;
import java.lang.*;
import java.io.*;
public final class ProblemD
{
    public static BufferedWriter out = new BufferedWriter(new OutputStreamWriter(System.out));
    public static int t, n, i, j, k, u, v, s, mod = (int)1e9+7;
    public static HashMap<Integer, Integer> count = new HashMap<>();
    public static TreeSet<Integer> set = new TreeSet<>(), count_helper = new TreeSet<>();
    public static void main(String[] args) throws IOException
    {
        FastScanner sc = new FastScanner();
        Print print = new Print();
        n = sc.nextInt();
        t = sc.nextInt();
        for(i = 0; i < n; i++)
            Add(sc.nextInt());
        print.integer(query());
        while(t-- > 0)
        {
            u = sc.nextInt();
            v = sc.nextInt();
            if(u == 1)
                Add(v);
            else
                Remove(v);
            print.integer(query());
        }
        out.close();
    }

    public static void Add(int value)
    {
        if(set.size() == 0)
        {
            set.add(value);
            return;
        }
        if(set.lower(value) == null)
        {
            count.put(set.first()-value, count.getOrDefault(set.first()-value, 0) + 1);
            count_helper.add(set.first()-value);
        }
        else if(set.higher(value) == null)
        {
            count.put(value - set.last(), count.getOrDefault(value - set.last(), 0) + 1);
            count_helper.add(value - set.last());
        }
        else
        {
            int t1 = set.lower(value), t2 = set.higher(value);
            count.put(t2 - t1, count.get(t2 - t1) - 1);
            if(count.get(t2 - t1) == 0) count_helper.remove(t2 - t1);
            count.put(value - t1, count.getOrDefault(value - t1, 0) + 1);
            count_helper.add(value - t1);
            count.put(t2 - value, count.getOrDefault(t2 - value, 0) + 1);
            count_helper.add(t2 - value);
        }
        set.add(value);
    }

    public static void Remove(int value)
    {
        set.remove(value);
        if(set.size() == 0) return;
        if(set.lower(value) == null)
        {
            count.put(set.first() - value, count.getOrDefault(set.first()-value, 0) - 1);
            if(count.get(set.first() - value) == 0) count_helper.remove(set.first() - value);
        }
        else if(set.higher(value) == null)
        {
            count.put(value - set.last(), count.getOrDefault(value-set.last(), 0) - 1);
            if(count.get(value - set.last()) == 0) count_helper.remove(value - set.last());
        }
        else
        {
            int t1 = set.lower(value), t2 = set.higher(value);
            count.put(value - t1, count.get(value - t1) - 1);
            if(count.get(value - t1) == 0) count_helper.remove(value - t1);
            count.put(t2 - value, count.get(t2 - value) - 1);
            if(count.get(t2 - value) == 0) count_helper.remove(t2 - value);
            count.put(t2 - t1, count.getOrDefault(t2 - t1, 0) + 1);
            count_helper.add(t2 - t1);
        }
    }

    public static int query()
    {
        if(set.size() <= 2)
            return 0;
        return set.last() - set.first() - count_helper.last();
    }

    public static class Print
    {
        Print()
        {}

        void integer(int... arr) throws IOException
        {
            for(int value: arr)
            {
                out.write(value+" ");
                out.flush();
            }
            out.write("\n");
            out.flush();
        }

        void string(String... arr) throws IOException
        {
            for(String value: arr)
            {
                out.write(value+" ");
                out.flush();
            }
            out.write("\n");
            out.flush();
        }

        void character(char... arr) throws IOException
        {
            for(char value: arr)
            {
                out.write(value+" ");
                out.flush();
            }
            out.write("\n");
            out.flush();
        }

        void long_int(long... arr) throws IOException
        {
            for(long value: arr)
            {
                out.write(value+" ");
                out.flush();
            }
            out.write("\n");
            out.flush();
        }
    }

    static class FastScanner
    {
        BufferedReader br=new BufferedReader(new InputStreamReader(System.in));
        StringTokenizer st=new StringTokenizer("");
        String next()
        {
            while (!st.hasMoreTokens())
                try
                {
                    st=new StringTokenizer(br.readLine());
                } catch (IOException e)
                {
                    e.printStackTrace();
                }
            return st.nextToken();
        }

        int nextInt()
        {
            return Integer.parseInt(next());
        }
        int[] readArray(int n)
        {
            int[] a=new int[n];
            for (int i=0; i<n; i++) a[i]=nextInt();
            return a;
        }
        long nextLong()
        {
            return Long.parseLong(next());
        }
    }
}