
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 $F\_1$ is the left focus of the ellipse $C: \frac{x^{2}}{2}+y^{2}=1$, and the line $l: y=x-1$ intersects the ellipse $C$ at points $A$ and $B$. Find the value of $|F\_1A|+|F\_1B|$.

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