
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
A set of 8 fixed points is given. For any arbitrarily chosen directed line $\iota$ on the plane, let the projections of $A_{1}, A_{2}, \cdots, A_{8}$ onto this line be $P_{1}, P_{2}, \cdots, P_{8}$. If these 8 projections do not coincide with each other and are ordered along the direction of line $\iota$ as $P_{i_{1}}, P_{i_{2}}, \cdots, P_{i_{8}}$, we obtain a permutation $i_{1}, i_{2}, \cdots, i_{8}$ of the integers $1, 2, \cdots, 8$. Define $N_{8}$ as the total number of distinct permutations obtained from the projections of these 8 points onto all directed lines on the plane, denoted as $N(A_{1}, A_{2}, \cdots, A_{8})$. Find the maximum value of $N_{8}$.

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