
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 $E$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ ($a>b>0$), a line passing through point $F_1(-c, 0)$ intersects the ellipse $E$ at points $A$ and $B$. If $|AF_1|=3|F_1B|$ and $AB\perp AF_2$, find the eccentricity of the ellipse $E$.

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