
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
A circle with center A has an arc $\overarc{BC}$ which is part of a circle centered at B, and similarly $\overarc{AC}$ is centered at C. Both centers B and C are endpoints of the arc, forming an equilateral triangle $\triangle ABC$. If the length of arc $\overarc{BC}$ is $15$ units, find the circumference of a smaller circle tangent to both arcs $\overarc{AC}$, $\overarc{BC}$, and line segment $\overline{BC}$.
A) $\frac{45}{4}$  
B) $\frac{90}{\pi}$  
C) $\frac{45}{2}$  
D) $\frac{90}{4}$  
E) $\frac{120}{\pi}$

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