
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
Given a positive integer \( k > 2 \), the sequence formed by \( k \)-polygonal numbers is \( 1, k, 3k-3, 6k-8, \cdots \). This is a second-order arithmetic sequence with a second-order common difference of \( k-2 \).

The sequence of Fermat numbers is \( 3=2^{2^{0}}+1, 5=2^{2^{1}}+1, 17=2^{2^{2}}+1, \cdots \).

Find all positive integers \( k \) such that there are common terms in the above two sequences.

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