
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 $F_{1}$ and $F_{2}$ are the left and right foci of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \left(a > 0, b > 0\right)$, point $P$ lies on the hyperbola, $PF_{1}\bot PF_{2}$, circle $O$: $x^{2}+y^{2}=\frac{9}{4}(a^{2}+b^{2})$, line $PF_{1}$ intersects circle $O$ at points $A$ and $B$, line $PF_{2}$ intersects circle $O$ at points $M$ and $N$. If the area of quadrilateral $AMBN$ is $9b^{2}$, calculate the eccentricity of the hyperbola $C$.

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