311061: CF1928E. Modular Sequence

You are given two integers $x$ and $y$. A sequence $a$ of length $n$ is called modular if $a_1=x$, and for all $1 < i \le n$ the value of $a_{i}$ is either $a_{i-1} + y$ or $a_{i-1} \bmod y$. Here $x \bmod y$ denotes the remainder from dividing $x$ by $y$.

Determine if there exists a modular sequence of length $n$ with the sum of its elements equal to $S$, and if it exists, find any such sequence.

Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 2 \cdot 10^4$). The description of the test cases follows.

The first and only line of each test case contains four integers $n$, $x$, $y$, and $s$ ($1 \le n \le 2 \cdot 10^5$, $0 \le x \le 2 \cdot 10^5$, $1 \le y \le 2 \cdot 10^5$, $0 \le s \le 2 \cdot 10^5$) — the length of the sequence, the parameters $x$ and $y$, and the required sum of the sequence elements.

The sum of $n$ over all test cases does not exceed $2 \cdot 10^5$, and also the sum of $s$ over all test cases does not exceed $2 \cdot 10^5$.

For each test case, if the desired sequence exists, output "Yes" on the first line (without quotes). Then, on the second line, output $n$ integers $a_1, a_2, \ldots, a_n$ separated by a space — the elements of the sequence $a$. If there are multiple suitable sequences, output any of them.

If the sequence does not exist, output "No" on a single line.

You can output each letter in any case (lowercase or uppercase). For example, the strings "yEs", "yes", "Yes", and "YES" will be accepted as a positive answer.

3
5 8 3 28
3 5 3 6
9 1 5 79

YES
8 11 2 2 5 
NO
NO

In the first example, the sequence $[8, 11, 2, 5, 2]$ satisfies the conditions. Thus, $a_1 = 8 = x$, $a_2 = 11 = a_1 + 3$, $a_3 = 2 = a_2 \bmod 3$, $a_4 = 5 = a_3 + 3$, $a_5 = 2 = a_4 \bmod 3$.

In the second example, the first element of the sequence should be equal to $5$, so the sequence $[2, 2, 2]$ is not suitable.