
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 two foci of an ellipse $C$. Let $AB$ be a chord of the ellipse passing through the point $F_{2}$. In the triangle $\triangle F_{1}AB$, the lengths are given as follows:
$$
|F_{1}A| = 3, \; |AB| = 4, \; |BF_{1}| = 5.
$$

Find $\tan \angle F_{2}F_{1}B$.

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