
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 $C:\dfrac{x^2}{8}+\dfrac{y^2}{4}=1$ with its left focus at $F_1$, and the line $l:y=x-2$ intersects the ellipse $C$ at points $A$ and $B$.

    (1) Find the length of segment $AB$;

    (2) Find the area of $\Delta ABF_1$.

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