
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
Let $P$ be a point on the hyperbola $C: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a,b > 0)$. Let $F_1$ and $F_2$ respectively be the left and right foci of $C$, and $PF_2 \perpendicular F_1F_2$, $PF_1$ intersects the $y$-axis at point $Q$, and $O$ is the origin. If the quadrilateral $OF_2PQ$ has an inscribed circle, then the eccentricity of $C$ is ______.

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