
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
The hyperbola $C$: $\frac{x^2}{a^2} - y^2 = 1\left(a > 0\right)$ has left and right foci $F_{1}$ and $F_{2}$, respectively. Point $A$ is a moving point on the left branch of $C$. Line $AF_{2}$ intersects the right branch of $C$ at point $B$, and $|AB| = 3a$. The perimeters of triangles $ABF_{1}$ and $BF_{1}F_{2}$ are equal. Find $|F_{1}F_{2}|$.

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