
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 $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\ (a > 0,b > 0)$, its left and right foci are $F_{1}$ and $F_{2}$ respectively. A line passing through $F_{2}$ intersects the right branch of hyperbola $C$ at points $P$ and $Q$. If $PQ$ is perpendicular to $PF_{1}$ and $|PQ|=|PF_{1}|$, find the eccentricity of the hyperbola $e$.

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