
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 a hyperbola $C$ defined by $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \ (a > 0,b > 0)$, the left and right foci are denoted by $F_1$ and $F_2$ respectively. Let $P$ be a point on hyperbola $C$ and $Q$ be a point on the asymptote of the hyperbola $C$, both $P$ and $Q$ lie in the first quadrant, and satisfy $\overrightarrow{QP} = \overrightarrow{PF_2}$ and $\overrightarrow{QF_1} \cdot \overrightarrow{QF_2} = 0$. Find the eccentricity of the hyperbola $C$.

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