
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 \( n \) points \( A_1, A_2, \ldots, A_n \) on a plane, where the distance between any two of them does not exceed 1, and a number \( a \) (where \( 0 \leq a \leq 1 \)). What is the maximum possible number \( N_2(a, n) \) of segments \( A_iA_j \) (where \( i, j = 1, 2, \ldots, n \)) with length \( \geq a \), if 

a) \( n = 4 \)

b) \( n = 5 \)

c) \( n = 6 \)

(The number \( N_2(a, n) \) will, of course, be different for different values of \( a \)).

For an arbitrary (possibly very large) number \( n \) of points on a plane, it is known that for all \( n > 4 \):
\[ N_2\left(\frac{\sqrt{2}}{2}, n\right) = \left\lfloor \frac{n^2}{3} \right\rfloor \]

where the square brackets denote the floor function.

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