
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 $C\_1$: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0)$ with left and right foci $F\_1$ and $F\_2$ respectively. A parabola, $C\_2$, has its vertex at the origin and its latus rectum passing through the foci of the hyperbola $C\_1$. If a point $P$ on hyperbola $C\_1$ intersects with parabola $C\_2$ such that $PF\_2 \perp F\_1F\_2$, then the eccentricity of hyperbola $C\_1$, $e =$ ___.

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