
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 the left focus of the ellipse $\frac{x^2}{2}+y^2=1$ is $F_{1}$, and the line $l:y=x-1$ intersects the ellipse $C$ at points $A$ and $B$.  
$(1)$ Find the length of segment $AB$;  
$(2)$ Find the area of $\triangle ABF_{1}$.

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