
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 an ellipse, respectively. A line $l$ passing through $F_{1}$ intersects the ellipse at two distinct points $A$ and $B$. Connect $AF_{2}$ and $BF_{2}$.

$(1)$ Find the perimeter of $\triangle ABF_{2}$;

$(2)$ If $AF_{2} \perp BF_{2}$, find the area of $\triangle ABF_{2}$.

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