
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 hyperbola ${x^2}-\frac{{y^2}}{2}=1$, a line $l$ passing through the right focus of the hyperbola intersects the hyperbola at points $A$ and $B$. If there exists a real number $\lambda$ such that $|AB|=\lambda$ and there are exactly $3$ lines that satisfy this condition, calculate $\lambda$.

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