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E - Add and Mex

Score : $500$ points

Problem Statement

You are given an integer sequence $A=(A_1,A_2,\ldots,A_N)$ of length $N$.

Perform the following operation $M$ times:

• For each $i\ (1\leq i \leq N)$, add $i$ to $A_i$. Then, find the minimum non-negative integer not contained in $A$.

Constraints

• $1\leq N,M \leq 2\times 10^5$
• $-10^9\leq A_i\leq 10^9$
• All values in the input are integers.

Input

The input is given from Standard Input in the following format:

$N$ $M$

$A_1$ $A_2$ $\ldots$ $A_N$

Output

Print $M$ lines.

The $i$-th $(1\leq i \leq M)$ line should contain the minimum non-negative integer not contained in $A$ after the $i$-th operation.

3 3
-1 -1 -6

2
2
0

The $1$-st operation makes the sequence $A$

$(-1 + 1, -1 +2 ,-6+3) = (0,1,-3).$

The minimum non-negative integer not contained in $A$ is $2$.

The $2$-nd operation makes the sequence $A$

$(0 + 1, 1 +2 ,-3+3) = (1,3,0).$

The minimum non-negative integer not contained in $A$ is $2$.

The $3$-rd operation makes the sequence $A$

$(1 + 1, 3 +2 ,0+3) = (2,5,3).$

The minimum non-negative integer not contained in $A$ is $0$.

5 6
-2 -2 -5 -7 -15

1
3
2
0
0
0