
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 \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a>0, b>0)\), its right focus is denoted as \(F\). A line \(l\) passing through \(F\) and perpendicular to the \(x\)-axis intersects the asymptotes at points \(A\) and \(B\). Point \(P\) is an intersection of line \(l\) with the hyperbola. Let \(O\) be the origin. If there are real numbers \(m\) and \(n\) such that \(\overrightarrow{O P}=m \overrightarrow{O A}+ n \overrightarrow{O B}\) and \(m n=\frac{2}{9}\), determine the eccentricity of the hyperbola.

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