
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 $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ with foci $F_1$ and $F_2$, the chord $AB$ passes through $F_1$ and lies on one branch of the hyperbola. If $|AF_{2}| + |BF_{2}| = 2|AB|$, find the value of $|AB|$.

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