
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 $F_1$ is the left focus of the hyperbola $\frac{x^{2}}{14} - \frac{y^{2}}{11} = 1$, and line $l$ passes through the origin intersecting hyperbola $C$ at points $P$ and $Q$. If $\overrightarrow{PF_1} \cdot \overrightarrow{QF_1} = 0$, find the perimeter of $\triangle PF_1Q$.

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