
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 two foci of the ellipse $E: \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$, a line $l$ passing through the origin intersects $E$ at points $A$ and $B$, and $\overrightarrow{AF_{2}} \cdot \overrightarrow{BF_{2}} = 0$, and $\frac{|\overrightarrow{AF_{2}}|}{|\overrightarrow{BF_{2}}|} = \frac{3}{4}$, calculate the eccentricity of $E$.

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