
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 tangent line drawn from the left focus $F(-c,0)$ of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ to the circle $x^2 + y^2 = \frac{a^2}{4}$, extended to intersect the right branch of the hyperbola at point $P$, and given that $\overrightarrow{OE} = \frac{1}{2}(\overrightarrow{OF} + \overrightarrow{OP})$, determine the eccentricity of the hyperbola.

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