
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
In  $\triangle ABC$ , the sides have integers lengths and  $AB=AC$ . Circle  $\omega$  has its center at the incenter of  $\triangle ABC$ . An *excircle* of  $\triangle ABC$  is a circle in the exterior of  $\triangle ABC$  that is tangent to one side of the triangle and tangent to the extensions of the other two sides. Suppose that the excircle tangent to  $\overline{BC}$  is internally tangent to  $\omega$ , and the other two excircles are both externally tangent to  $\omega$ . Find the minimum possible value of the perimeter of  $\triangle ABC$ .

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