
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
Let $F_1$ and $F_2$ be the left and right foci of the ellipse $E$: $\dfrac{x^{2}}{a^{2}}+ \dfrac{y^{2}}{b^{2}}=1(a > b > 0)$, respectively. A line passing through point $F_1$ intersects the ellipse $E$ at points $A$ and $B$, with $|AF_1|=3|F_1B|$ and $|AB|=4$. The perimeter of $\triangle ABF_2$ is $16$. Find $|AF_2|$.

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