
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
A hyperbola $C$ has left and right foci at $F\_1(-1,0)$ and $F\_2(1,0)$, respectively. A parabola $y^{2}=4x$ intersects with the hyperbola $C$ at point $P$. If $(\overrightarrow{F\_2P}+\overrightarrow{F\_2F\_1}) \cdot (\overrightarrow{F\_2P}-\overrightarrow{F\_2F\_1})=0$, then the eccentricity of $C$ is $\_\_\_\_\_\_\_\_\_.$

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