
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]


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Given the hyperbola $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1(a>0,b>0)$, whose left and right foci are ${{F}_{1}}$ and ${{F}_{2}}$, respectively. A line passing through ${{F}_{1}}$, perpendicular to the $x$-axis, intersects the left branch of the hyperbola at points $A$ and $B$. If $\mathrm{\Delta }A{{F}_{2}}B$ is an acute triangle, find the eccentricity of the hyperbola.

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