P3179 [HAOI2015] Array Game

There is an array of length $n$. Two players, A and B, play the following game on it: initially, some cells in the array are white and some are black. The players take turns. On each turn, the current player chooses a white cell with index $x$. Then they choose an integer $k$ between $1 \ldots \lfloor \dfrac{n}{x} \rfloor$, and flip the colors of all cells with indices $x, 2\times x, \ldots, k\times x$. The player who cannot make a move loses. Now A (the first player) has several queries. For each query, you are given an initial state of the array. You need to determine whether A has a winning strategy for that initial state.

The first line contains an integer $n$, the length of the array. The second line contains an integer $k$, the number of queries. The next $2\times k$ lines describe the queries, two lines per query. In these two lines, the first line contains a positive integer $w$, the number of white cells in the array. The second line contains $w$ integers between $1$ and $n$, inclusive, which are the indices of the white cells.

For each query, output one line with a string: output `Yes` if the first player has a winning strategy, otherwise output `No`.

#### Explanation for Sample Input/Output 1 In the first query, A chooses index $1$ and flips cells $1\times 1$ and $2\times 1$. In the second query, no matter which index A chooses, they can flip only one cell. B can then flip the other cell. #### Constraints For $30\%$ of the testdata, $n \leq 20$. For $50\%$ of the testdata, $n \leq 10^6$. For $70\%$ of the testdata, $n \leq 10^7$. For $100\%$ of the testdata, $n \leq 10^9$, $k, w \leq 100$, and no cell appears more than once within a single query. Translated by ChatGPT 5