
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
There are $n$ matches on the table ($n > 1$). Two players take turns removing matches from the table. The first player removes any number of matches between 1 and $n-1$, and each subsequent player can remove no more matches than the number taken by their partner on the previous turn. The player who takes the last match wins. Find all $n$ for which the first player can guarantee a win.

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