
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 length of the imaginary axis of a hyperbola is $4$, its eccentricity $e= \frac{ \sqrt{6}}{2}$, and $F\_1$, $F\_2$ are its left and right foci respectively, find the value of $|AB|$, where $AB$ is the line segment formed by the intersection of a line passing through $F\_1$ with the left branch of the hyperbola, given that $|AB|$ is the arithmetic mean of $|AF\_2|$ and $|BF\_2|$.

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