
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 \( n \) fleas on an infinite sheet of triangulated paper. Initially, the fleas are in different small triangles, all of which are inside an equilateral triangle consisting of \( n^{2} \) small triangles. Once a second, each flea jumps from its triangle to one of the three small triangles as indicated. For which positive integers \( n \) does there exist an initial configuration such that after a finite number of jumps all the \( n \) fleas can meet in a single small triangle?

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