
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 8 arbitrarily chosen points \( A_{1}, A_{2}, \ldots, A_{8} \) in the plane, consider any directed line \( l \) on the plane. Let the projections of \( A_{1}, A_{2}, \ldots, A_{8} \) onto this line be \( P_{1}, P_{2}, \ldots, P_{8} \). If these 8 projections are distinct and their order along the direction of line \( l \) is \( P_{i_{1}}, P_{i_{2}}, \ldots, P_{i_{8}} \), this gives a permutation \( i_{1}, i_{2}, \ldots, i_{8} \) of \( 1, 2, \ldots, 8 \). Let \( N_{8}=N(A_{1}, A_{2}, \ldots, A_{8}) \) be the number of different permutations obtained by projecting the 8 points onto all possible directed lines. Determine the maximum value of \( N_{8} \).

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