
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 $F_{1}(-c,0)$ and $F_{2}(c,0)$ are the two foci of the hyperbola $C: \frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1(a \gt 0, b \gt 0)$, $P$ is a point on the hyperbola $C$, $PF_{1} \perp PF_{2}$, and $∠PF_2F_1=\frac{π}{3}$, find the eccentricity of the hyperbola $C$.

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