
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 that $F_1$ and $F_2$ are the left and right foci of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ $(a>0,b>0)$, there exists a point $P$ on the hyperbola such that $|PF_1|+|PF_2|=3b$ and $|PF_1|\cdot|PF_2|=\frac{9}{4}ab$. Determine the eccentricity of the hyperbola.

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