
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 $\triangle ABC$, $AB=AC$, and the sides have integer lengths. A circle $\omega$, centered at the incenter of $\triangle ABC$, has different tangent properties with the excircles: an excircle tangent to $BC$ is internally tangent to $\omega$, while the other two excircles are internally tangent too. Determine the smallest possible perimeter of $\triangle ABC$ under these conditions.

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