
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 ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with $a > b > 0$, the left and right foci are $F_{1}$ and $F_{2}$, respectively. A line $l$ passing through $F_{1}$ intersects the ellipse at points $A$ and $B$, and the y-intercept of $l$ is $1$. If $|AF_{1}| = 3|F_{1}B|$ and $AF_{2}$ is perpendicular to the x-axis, find the length of the major axis of this ellipse.

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