
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 the eccentricity of the hyperbola $C$ is $2$, with left and right foci $F\_1$ and $F\_2$ respectively, and point $A$ is on $C$. If $|F\_1A|=2|F\_2A|$, then $\cos ∠AF\_2F\_1=$ _____ .

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