P17301 [ICPC 2026 Xi'an I] Unreachable Land

Yuki dreams of reaching the unreachable land. After years of hard work, only this problem remains before her. Given three integers $a, b, m$. You need to perform $m$ rounds of operations. In the $i$-th round, you can either set $a \leftarrow a \bmod (m - i + 1)$ or choose not to modify $a$. Find the number of ways to make $a = b$ after $m$ rounds, modulo $998244353$. Two schemes are considered different if and only if there exists some $1 \le i \le m$ such that in one scheme you performed a modification in the $i$-th round, while in the other you did not. Note that choosing to perform $a \leftarrow a \bmod (m - i + 1)$ is considered a modification, regardless of whether the value of $a$ changes after the operation. You once dreamed of reaching the unreachable land that only exists in fairy tales. Now that Yuki has a chance to realize this dream, you must help her.

This problem contains multiple test cases. The first line contains a positive integer $t$ $(1 \le t \le 10^5)$, representing the number of test cases. For each test case: - A single line contains three integers $a, b, m$ $(0 \le b < m \le a \le 2 \cdot 10^5)$. It is guaranteed that the sum of $a$ over all test cases does not exceed $2 \cdot 10^5$.

For each test case, output a single line containing an integer representing the answer modulo $998244353$.

For the first test case: - One valid operation scheme is to perform modifications in the $3$-rd and $4$-th rounds. - Another valid operation scheme is to perform modifications in all rounds from $1$ to $5$. For the second test case: - The only valid operation scheme is to perform a modification in the $3$-rd round. For the fourth test case: - It can be proven that no valid operation scheme exists.