CF1764H.Doremy's Paint 2

Doremy has $n$ buckets of paint which is represented by an array $a$ of length $n$ . Bucket $i$ contains paint with color $a_i$ . Initially, $a_i=i$ .

Doremy has $m$ segments $[l_i,r_i]$ ( $1 \le l_i \le r_i \le n$ ). Each segment describes an operation. Operation $i$ is performed as follows:

• For each $j$ such that $l_i < j \leq r_i$ , set $a_j := a_{l_i}$ .

Doremy also selects an integer $k$ . She wants to know for each integer $x$ from $0$ to $m-1$ , the number of distinct colors in the array after performing operations $x \bmod m +1, (x+1) \bmod m + 1, \ldots, (x+k-1) \bmod m +1$ . Can you help her calculate these values? Note that for each $x$ individually we start from the initial array and perform only the given $k$ operations in the given order.

The first line of input contains three integers $n$ , $m$ , and $k$ ( $1\le n,m\le 2\cdot 10^5$ , $1 \le k \le m$ ) — the length of the array $a$ , the total number of operations, and the integer that Doremy selects.

The $i$ -th line of the following $m$ lines contains two integers $l_i$ , $r_i$ ( $1\le l_i\le r_i\le n$ ) — the bounds of the $i$ -th segment.

Output $m$ integers. The $(x+1)$ -th integer should be the number of distinct colors in the array if we start from the initial array and perform operations $x \bmod m +1, (x+1) \bmod m + 1, \ldots, (x+k-1) \bmod m +1$ .

In the first test case, the picture below shows the resulting array for the values of $x=0,1,2$ respectively.