
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 the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ with its right focus at $F(\sqrt{5},0)$, points $P$ and $Q$ lie on the hyperbola and are symmetric with respect to the origin $O$. If $PF\perp QF$ and the area of $\triangle PQF$ is $4$, then the eccentricity of the hyperbola is $e=$____.

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