
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 focus $F(-c, 0)$ intersects the circle $x^{2} + y^{2} = a^{2}$ at a tangent point $E$. Extend $FE$ to intersect the hyperbola at point $P$, and let $O$ be the origin. If $\overrightarrow{OE} = \frac{1}{2}(\overrightarrow{OF} + \overrightarrow{OP})$, find the eccentricity of the hyperbola.

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