
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
Let \( A_{1}, A_{2}, \cdots, A_{8} \) be any 8 points taken on a plane. For any directed line \( l \) on the plane, let the projections of \( A_{1}, A_{2}, \cdots, A_{8} \) on that line be \( P_{1}, P_{2}, \cdots, P_{8} \) respectively. If the 8 projections do not coincide pairwise, they are arranged in the direction of line \( l \) as \( P_{i_{1}}, P_{i_{2}}, \cdots, P_{i_{8}} \). In this way, we get a permutation \( i_{1}, i_{2}, \cdots, i_{8} \) of \( 1, 2, \cdots, 8 \) (in the picture, this permutation is \( 2, 1, 8, 3, 7, 4, 6, 5 \)). Let the number of different permutations obtained after projecting these 8 points onto all directed lines on the plane be \( N_{8} = N\left(A_{1}, A_{2}, \cdots, A_{8}\right) \). Find the maximum value of \( N_{8} \).

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