
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
Consider the hyperbola given by the equation $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0)$. Let the right focus $F$ be the foot of the perpendicular drawn from the asymptote's vertical line. Denote the foot of this perpendicular as $P$ (which lies in the first quadrant), and extend $FP$ to intersect the parabola $y^2 = 2px (p > 0)$ at point $Q$. Given that the hyperbola and the parabola share a common focus, and that $\overrightarrow{OP} = \frac{1}{2} (\overrightarrow{OF} + \overrightarrow{OQ})$, find the square of the eccentricity of the hyperbola.

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