
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 $\triangle ABC$, $AB = BC$, and sides have integer lengths. A circle $\omega$ has its center at the incenter of $\triangle ABC$. There are three excircles corresponding to the sides of the triangle. The excircle tangent to side $AC$ is internally tangent to $\omega$. The excircles corresponding to sides $AB$ and $BC$ are externally tangent to $\omega$. Another smaller circle $\phi$ is externally tangent to the excircle of side $AB$ and internally 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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