
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 right 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 and passing through point $F$ intersects the two asymptotes at points $A$ and $B$, and intersects one of the branches of the hyperbola at point $P$. Let the origin be $O$. If $\overrightarrow{OP} = n \overrightarrow{OA} + n \overrightarrow{OB} (m, n \in \mathbb{R})$, and $mn = \dfrac{2}{9}$, then the eccentricity of the hyperbola is $\boxed{\dfrac{3\sqrt{2}}{4}}$.

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