
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 right focus $F\_2$ of the ellipse $C$: $\frac{x^{2}}{4} + \frac{y^{2}}{3} = 1$, it intersects with the ellipse $C$ at points $A$ and $B$. If $\overrightarrow{AF\_2} = \overrightarrow{F\_2B}$, find the distance between point $A$ and the left focus $F\_1$ ($|AF\_1| = \_\_\_\_\_\_$).

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