Description

Vinsta and Stella have $n$ piles of stones, and the $i$-th pile has $s_i$ stones.

They agree to take turns starting from Vinsta. In each move, one may choose at least $1$ pile and at most $k$ piles. For the $i$-th pile, one may choose two real numbers $a, b$ such that:

• $a \times b = s_i$
• $a + b = c, c \in [1, s_i] \cap \Z$

Then discard $c$ stones from the $i$-th pile, i.e., $s_i \leftarrow s_i - c$. The player who cannot make a move loses. They want to know whether Vinsta has a winning strategy.

Input Format

The first line contains a positive integer $T$, the number of test cases.

Then follow $T$ test cases. For each test case, the first line contains three positive integers $n, k, S$, where $S = \max\{s_i\}$.

The second line contains $n$ positive integers $s_i$, representing the initial number of stones in the $i$-th pile.

Output Format

For each test case, output one line. If there is a winning strategy, output YES, otherwise output NO.

2
7 1 13
2 3 4 5 7 10 11
8 1 13
2 3 4 5 7 10 11 13

YES
NO

1
7 2 100
19 26 8 17 11 45 14

YES

Hint

For $100\%$ of the testdata, $1 \le T \le 10$, $1 \le k \le n \le 3 \times 10^6$, $1 \le S \le 3 \times 10^{17}$.

Translated by ChatGPT 5