
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
The ancient Greek mathematician Ptolemy gave Ptolemy's theorem in his famous work "Mathematical Compilation," which states that in any convex quadrilateral, the product of the two diagonals is less than or equal to the sum of the products of the two pairs of opposite sides, and equality holds if and only if the four vertices of the convex quadrilateral lie on a circle. Given that the left and right foci of the hyperbola $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1\left(a \gt 0,b \gt 0\right)$ are $F_{1}$ and $F_{2}$ respectively, and two points $A$ and $B$ symmetric with respect to the origin on the hyperbola satisfy $|AB|\cdot |F_{1}F_{2}|=|AF_{1}|\cdot |BF_{2}|+|AF_{2}|\cdot |BF_{1}|$, if $\angle AF_{1}F_{2}=\frac{π}{6}$, then the eccentricity of the hyperbola is ______.

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