
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 $\triangle ABC$, with $AB=AC$, let sides $AB$ and $AC$ be twice the length of side $BC$. Suppose $\triangle ABC$ contains a circle $\omega$, centered at the incenter of $\triangle ABC$. There are three excircles each tangent to one side of $\triangle AB C$ and the extensions of the other two sides. Assume the excircle tangent to $\overline{BC}$ is internally tangent to $\omega$, and the other two excircles are externally tangent to $\omega$. Determine the minimum possible value of the perimeter of $\triangle ABC$ assuming all side lengths are integers.

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