
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 hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0, b > 0$), let F be the right focus of the hyperbola. A perpendicular line from point F to the x-axis intersects the two asymptotes at points A and B, and intersects the hyperbola in the first quadrant at point P. Let O be the origin of the coordinate system. If $\vec{OP} = \lambda \vec{OA} + \mu \vec{OB}$ ($\lambda, \mu \in \mathbb{R}$), and $\lambda^2 + \mu^2 = \frac{5}{8}$, calculate the eccentricity of the hyperbola.

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