
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
For the hyperbola \( x^{2} - \frac{y^{2}}{2} = 1 \), a line \( l \) passes through its right focus and intersects the hyperbola at points \( A \) and \( B \). If there exists a real number \(\lambda\) such that the distance \( |AB| = \lambda \) and there are exactly 3 distinct lines \( l \) satisfying this condition, find the value of \(\lambda\).

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