
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 a table ($n > 1$). Two players take turns removing the matches from the table. On the first move, a player can remove any number of matches from 1 to $n-1$. After that, each player can only remove up to as many matches as the previous player took on their move. 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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