
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
Consider a hyperbola with the equation $x^2 - y^2 = 9$. A line passing through the left focus $F_1$ of the hyperbola intersects the left branch of the hyperbola at points $P$ and $Q$. Let $F_2$ be the right focus of the hyperbola. If the length of segment $PQ$ is 7, then calculate the perimeter of $\triangle F_2PQ$.

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