
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 an isosceles triangle $\triangle ABC$ with $AB = AC$, an excircle of $\triangle ABC$ is tangent to side $\overline{BC}$ at a point $D$. Suppose furthermore that the circle $\omega$, centered at the incenter of $\triangle ABC$, is internally tangent to this excircle. If $C$ is at coordinates $(0,0)$ and $B$ is at $(s,0)$, with $s$ being the length of side $BC$, determine the minimum possible perimeter of $\triangle ABC$.

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