
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 $F(-c,0)$ of the hyperbola $\frac {x^{2}}{a^{2}} - \frac {y^{2}}{b^{2}} = 1$ ($a > 0, b > 0$) to the circle $x^{2} + y^{2} = a^{2}$. Let the point of tangency be $E$. Extend $FE$ to intersect the parabola $y^{2} = 4cx$ at point $P$, and let $O$ be the origin. If $\overrightarrow{OE} = \frac {1}{2}(\overrightarrow{OF} + \overrightarrow{OP})$, then the eccentricity of the hyperbola is \_\_\_\_\_\_.

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