
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
Given that $F$ is the left focus of the hyperbola $C: \frac{x^{2}}{9} - \frac{y^{2}}{16} = 1$, and $P, Q$ are points on the right branch of $C$. If the length of $PQ$ is twice the length of the imaginary axis, and point $A(5, 0)$ is on the line segment $PQ$, calculate the perimeter of $\triangle PFQ$.

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