
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 the hyperbola $x^{2}- \frac{y^{2}}{4}=1$. Let $F\_1$ be its left focus and let a line $l$ intersect the left branch of the hyperbola at points $P$ and $Q$. If $|PQ|=4$ and $F\_2$ is the right focus of the hyperbola, then the perimeter of triangle $PF\_2Q$ is \_\_\_\_\_\_.

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