
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Consider a two-player turn-based game. Initially, there are $k$ piles of stones, with the number of stones in each pile given by $n_{1}, n_{2}, \ldots, n_{k}$. In each turn, a player is allowed to take 1 or more stones from one pile, and then they can distribute some of the remaining stones to other piles (piles reduced to zero cannot have stones added back to them). The player who performs the last move wins. For what array $\left(n_{1}, n_{2}, \ldots, n_{k}\right)$ does the first player have a winning strategy?

Present the answer in LaTex format: \boxed{Your answer}
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