
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 that $F_1$ and $F_2$ are the left and right foci of the ellipse $E$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1 (a > b > 0)$, and the line passing through point $F_1$ intersects ellipse $E$ at points $A$ and $B$, with $|AF_1|=3|BF_1|$.

(1) If $|AB|=4$ and the perimeter of $\triangle ABF_2$ is $16$, find $|AF_2|$.

(2) If $\cos \angle AF_2B=\frac{3}{5}$, find the eccentricity of ellipse $E$.

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