
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 two foci of the hyperbola $\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1 (a > 0, b > 0)$, and $P$ is a point on the hyperbola, if $\overrightarrow{P{{F}\_{1}}} \cdot \overrightarrow{P{{F}\_{2}}}=0$ and $|\overrightarrow{P{{F}\_{1}}}| \cdot |\overrightarrow{P{{F}\_{2}}}|=2ac$, determine the eccentricity of the hyperbola.

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