
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
Let \( C \) be the unit circle. Four distinct, smaller congruent circles \( C_{1}, C_{2}, C_{3}, C_{4} \) are internally tangent to \( C \) such that \( C_{i} \) is externally tangent to \( C_{i-1} \) and \( C_{i+1} \) for \( i = 1, \ldots, 4 \), where \( C_{5} \) denotes \( C_{1} \) and \( C_{0} \) represents \( C_{4} \). Compute the radius of \( C_{1} \).

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