| **Note: The only difference between this and [Chapter 1](https://www.facebook.com/codingcompetitions/hacker-cup/2022/round-3/problems/D1/) is that here, you must find the sum of answers for \(K = 1..N\).** | |
| Boss Rob ain’t Som Tawyer, but he can paint a fence alright. | |
| Rob's fence is made of \(N\) wooden stakes, numbered \(1\) to \(N\) from left to right. Initially (at time \(0\)), the \(i\)th stake is of color \(i\). There is a fencepost before stake \(1\) and after stake \(N\), as well as after every \(K\)th stake starting from the left. | |
| Rob has a simple and joyful plan to repaint his fence, consisting of \(M\) moments in time. At time \(i\), he'll repaint all stakes which are color \(A_i\) to color \(B_i\). Doing so, when would be the *first time* that all pairs of stakes not separated by a fencepost have the same color? If it will never occur, consider the answer to be \(-1\). | |
| Rob is still on the _fence_ about the value of \(K\), so please print **the sum of answers over \(K = 1..N\).** Sorry for the pun (we know there is a lot at _stake_ in this round _gating_ the finals). In our de_fence_, it makes for good _post_-problem content. | |
| # Constraints | |
| \(1 \le T \le 30\) | |
| \(1 \le N, M \le 600{,}000\) | |
| \(1 \le A_i, B_i \le N\) | |
| \(A_i \ne B_i\) | |
| The sum of \(N\) across all test cases is at most \(4{,}000{,}000\). | |
| The sum of \(M\) across all test cases is at most \(4{,}000{,}000\). | |
| # Input Format | |
| Input begins with an integer \(T\), the number of test cases. For the \(i\)th test case, there is first a line containing two space-separated integers \(N\) and \(M\). Then, \(M\) lines follow, the \(i\)th of which contains two space-separated integers \(A_i\) and \(B_i\). | |
| # Output Format | |
| For the \(i\)th test case, output a single line containing `"Case #i: "` followed by a single integer, the sum of answers for each \(K = 1..N\). | |
| # Sample Explanation | |
| The progressions of fences for the first two sample cases are depicted below (with fenceposts depicted for \(K=2\) and \(K=3\) respectively): | |
| {{PHOTO_ID:631752615085471|WIDTH:700}} | |
| In the first case, the answers are \([0, 2, 4, 3, 4]\) for \(K = 1..5\), so we print \(2+4+3+4 = 13\). | |
| In the second case, the answers are \([0, 1, 3]\) for \(K = 1..3\), so we print \(1+3=4\). | |
| In the third case, the fence (with fenceposts depicted for \(K=3\)) progresses as follows: | |
| {{PHOTO_ID:413921204256250|WIDTH:700}} | |
| For \(K = 1..8\), the answers are \([0, 5, -1, 6, -1, -1, -1, -1]\), so we print the sum \(6\). | |