
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
Let the left focus of the hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1 (a > 0, b > 0)$ be $F$. A line $l$ perpendicular to the $x$-axis passes through point $F$ and intersects the two asymptotes at points $M$ and $N$, and intersects the hyperbola in the second quadrant at point $P$. Let $O$ be the origin. If $\overrightarrow{OP} = m\overrightarrow{OM} + n\overrightarrow{ON} (m, n \in \mathbf{R})$, and $mn = \dfrac{1}{8}$, then the eccentricity of the hyperbola is $\boxed{\sqrt{2}}$.

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