Doremy's Paint 2

给定一个长度为 $n$ 的序列 $a_{1\cdots n}$，初始有 $a_i=i$。 现在有 $m$ 个操作，第 $i$ 个操作有参数 $l_i,r_i(l_i\le r_i)$，执行该操作会把 $a_{l_i+1\cdots r_i}$ 赋值为 $a_l$。 给定 $k$，对于所有 $x=1\sim m$，求如果依次执行第 $x$ 个到第 $x+k-1$ 个操作那么序列中有多少种不同的元素。这里将第 $i+m$ 个操作视为第 $i$ 个操作。

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 $ .

7 5 5
3 5
2 3
4 6
5 7
1 2

3 3 3 3 2

10 9 4
1 1
2 3
3 4
7 9
6 8
5 7
2 4
9 10
1 3

6 6 7 6 5 5 7 7 7

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