
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 $\triangle PQR$, the sides have integer lengths, and $PQ=PR$. Circle $\sigma$ has its center at the incenter of $\triangle PQR$. An excircle of $\triangle PQR$ is a circle in the exterior of $\triangle PQR$ that is tangent to one side of the triangle and tangent to the extensions of the other two sides. Suppose that the excircle tangent to $\overline{QR}$ is internally tangent to $\sigma$, and the other two excircles are externally tangent to $\sigma$. Also, suppose that $\angle QRP = 120^\circ$. Find the minimum possible value of the perimeter of $\triangle PQR$.

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