
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 $n$ pockets, each containing $k_{1}, k_{2}, \cdots, k_{n}$ balls. Two players, A and B, take turns to remove balls. In each turn, a player can take at least 1 and at most $m$ balls from one pocket. The player who takes the last ball wins. This problem is denoted as $\left\{k_{1}, k_{2}, \cdots, k_{n}\right\}_{m}$. For the problem $\{1,2,3, \cdots, n\}_{3}$, for what values of $n$ does player A (the first player) have a winning strategy?

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