
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
The coins \( F_{1}, F_{2}, \ldots, F_{n} \) are one-forint coins fixed on a plane such that each coin \( F_{i} \) touches the coin \( F_{i+1} \) (for \( i = 1, 2, \ldots, n \), and \( F_{n+1} \) means \( F_{1} \)). Additionally, a similar coin \( G \) can roll externally around this "wreath," touching each \( F_{i} \)'s arc in succession. How many rotations does \( G \) make around its own center before it first returns to its starting position?

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