
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 left focus $F(-c,0)$ of the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ $(a > 0,b > 0)$, draw a circle with equation $x^{2}+y^{2}=\frac{a^{2}}{4}$. Let $E$ be the point of intersection between the circle and the tangent line from $F$. Extend $FE$ to intersect the right branch of the hyperbola at $P$. If $\overrightarrow{OP}=2\overrightarrow{OE}-\overrightarrow{OF}$, determine the eccentricity of the hyperbola.

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