
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 tangent line is drawn from the left focus $F(-c,0)$ ($c > 0$) of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0$, $b > 0$) to the circle $x^2 + y^2 = \frac{a^2}{4}$. Let the point of tangency be $E$. Extend $FE$ to intersect the right branch of the hyperbola at point $P$. If $\overrightarrow{OP} = 2\overrightarrow{OE} - \overrightarrow{OF}$, then the eccentricity of the hyperbola is $\boxed{\frac{\sqrt{10}}{2}}$.

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