
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 $C:\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1(a＞0,b＞0)$, $P$ is a point on the hyperbola in the first quadrant, and $∠{F}_{1}P{F}_{2}=\frac{π}{3}$, $|{F}_{1}{F}_{2}|=2\sqrt{3}$, the symmetric point $Q$ of $F_{1}$ with respect to the bisector of $\angle F_{1}PF_{2}$ lies exactly on $C$. Then the eccentricity of $C$ is ______.

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