
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 the diagram, five circles of radius 1 with centers $P$, $Q$, $R$, $S$, and $T$ are tangent to one another and to the sides of $\triangle ABC$. Circle $P$ is tangent to $AB$ and $AC$, circle $Q$ is tangent to $AB$ and $BC$, circle $R$ is tangent to $BC$, circle $S$ is tangent to $BC$ and $CA$, and circle $T$ is tangent to $CA$. The centers form a closed chain $PQRS$ and $T$ is inside $\triangle ABC$. $\triangle ABC$ is an isosceles triangle with $AB = AC$. Find the perimeter of $\triangle ABC$.

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