
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]


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A tile, in the shape of a regular polygon, was removed from its place in a panel. It was observed that if this tile were rotated by \( 40^\circ \) or \( 60^\circ \) around its center, it could fit perfectly into the spot that was left vacant on the panel. What is the smallest number of sides that this polygon can have?

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