CF2247B.Yet Another Constructive

You are given three integers $n$, $k$, and $m$.

We call an array $a$ of length $n$, consisting of positive integers, good if the minimum length of a non-empty subarray$^{\text{∗}}$ of $a$ whose sum is divisible by $m$ equals $k$.

Formally, an array $a$ is good if the following conditions hold:

• there exist integers $l$ and $r$ such that $1 \le l \le r \le n$, $r - l + 1 = k$, and $\sum\limits_{i = l}^{r} a_i$ is divisible by $m$;
• there do not exist integers $l$ and $r$ such that $1 \le l \le r \le n$, $r - l + 1 \lt k$, and $\sum\limits_{i = l}^{r} a_i$ is divisible by $m$.

Find any good array $a$, or determine that no such array exists.

$^{\text{∗}}$An array $a$ is a subarray of an array $b$ if $a$ can be obtained from $b$ by the deletion of several (possibly, zero or all) elements from the beginning and several (possibly, zero or all) elements from the end.

给你三个整数 $n$、$k$ 和 $m$。

我们称一个长度为 $n$、由正整数组成的数组 $a$ 是好数组，当且仅当：其所有非空子数组中，和能被 $m$ 整除的最短子数组的长度恰好等于 $k$。

形式化地，数组 $a$ 是好数组当且仅当满足以下两个条件：

• 存在整数 $l$ 和 $r$，使得 $1 \le l \le r \le n$，$r - l + 1 = k$，且 $\sum\limits_{i = l}^{r} a_i$ 能被 $m$ 整除；
• 不存在整数 $l$ 和 $r$，使得 $1 \le l \le r \le n$，$r - l + 1 \lt k$，且 $\sum\limits_{i = l}^{r} a_i$ 能被 $m$ 整除。

请构造任意一个好数组 $a$，或判断不存在这样的数组。

$^{\text{∗}}$ 若数组 $a$ 可通过从数组 $b$ 的开头删除若干（可能为零个或全部）元素，并从结尾删除若干（可能为零个或全部）元素而得到，则称 $a$ 是 $b$ 的一个子数组。

Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 10^4$). The description of the test cases follows.

The only line of each test case contains three integers $n$, $k$, and $m$ ($1 \le k \le n \le 2 \cdot 10^5$, $1 \le m \le 10^9$).

It is guaranteed that the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$.

每个测试包含多个测试用例。第一行包含测试用例的数量 $t$（$1 \le t \le 10^4$）。随后是测试用例的描述。

每个测试用例仅有一行，包含三个整数 $n$、$k$ 和 $m$（$1 \le k \le n \le 2 \cdot 10^5$，$1 \le m \le 10^9$）。

保证所有测试用例的 $n$ 之和不超过 $2 \cdot 10^5$。

For each test case, if an answer exists, output "YES" on the first line. On the second line, output $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^{18}$) — the array $a$ satisfying the conditions of the problem. If there are multiple valid solutions, output any of them.

Otherwise, output a single line containing "NO".

You can output the answer in any case (upper or lower). For example, the strings "yEs", "yes", "Yes", and "YES" will be recognized as positive responses.

对于每个测试用例，如果存在答案，则在第一行输出 “YES”。在第二行输出 $n$ 个整数 $a_1, a_2, \ldots, a_n$（$1 \le a_i \le 10^{18}$）——即满足题目条件的数组 $a$。若存在多个合法解，输出任意一个即可。

否则，输出一行，内容为 “NO”。

你可以以任意大小写形式输出答案（例如字符串 “yEs”、“yes”、“Yes” 和 “YES” 均会被识别为肯定回答）。

In the first example, one possible solution is $a = [1]$. The only non-empty subarray has length $k = 1$, and its sum is divisible by $m = 1$.

In the second example, one possible solution is $a = [9, 17, 14, 23, 11]$. The sum of the subarray $[9, 17, 14]$ is $9 + 17 + 14 = 40$, which is divisible by $m = 5$. It can be shown that no non-empty subarray with fewer than $k = 3$ elements has a sum divisible by $5$, so this solution is valid.

In the fourth example, it can be shown that no good array $a$ exists.

在第一个例子中，一个可能的解是 $a = [1]$。唯一的非空子数组长度为 $k = 1$，其和可被 $m = 1$ 整除。

在第二个例子中，一个可能的解是 $a = [9, 17, 14, 23, 11]$。子数组 $[9, 17, 14]$ 的和为 $9 + 17 + 14 = 40$，可被 $m = 5$ 整除。可以证明：不存在长度少于 $k = 3$ 的非空子数组，其和能被 $5$ 整除，因此该解有效。

在第四个例子中，可以证明不存在满足条件的好数组 $a$。