
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 \( F \) is the right focus of the hyperbola \( x^{2} - y^{2} = 1 \), \( l \) is the right directrix of the hyperbola, and \( A \) and \( B \) are two moving points on the right branch of the hyperbola such that \( A F \perp B F \). The projection of the midpoint \( M \) of line segment \( AB \) onto \( l \) is \( N \). Find the maximum value of \( \frac{|MN|}{|AB|} \).

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