
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 point $F$ is the right focus of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1(a>0,b>0)$, a line $l$ perpendicular to the $x$-axis is drawn through $F$, intersecting the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1(a>0,b>0)$ at points $P$ and $Q$. If $\triangle POQ$ is a right triangle with $O$ as the origin, determine the eccentricity of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1(a>0,b>0)$.

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