
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 $C$: $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1\left(a \gt 0,b \gt 0\right)$, and $P$ is a point on the right branch of the hyperbola. Find the eccentricity of the hyperbola given that $\angle F_{1}PF_{2}=60^{\circ}$ and the area of $\triangle{F}_{1}PF_{2}$ is $\sqrt{3}ac$.

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