
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 two circular arcs, \(AC\) and \(BC\), be centered at \(B\) and \(A\) respectively, with \(AB\) as the common chord. Suppose there is a third arc \(AD\) centered at \(C\) such that \(AC\) is subtended at \(C\), and each arc makes an angle of \(60^\circ\) each at their respective centers. If the length of \(BC\) is \(10\), find the circumference of the smallest circle tangent to all three arcs.
A) 55
B) 58
C) 60
D) 62
E) 65

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