
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 the hyperbola $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1 (a > 0, b > 0)$, let $A, B$ be the left and right foci of the hyperbola, respectively. If point $M$ is on the hyperbola and satisfies that $\Delta ABM$ is an isosceles triangle with a vertex angle of ${120}^{0}$, then the eccentricity of the hyperbola is $\_\_\_\_\_\_\_\_\_\_\_$.

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