
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
Nikees has a set  $S$  of  $n$  points on a plane and decides to colour them. All  $\dbinom{n}{2}$  line segments are drawn and they have distinct lengths. Find the maximum number of colours that are used at least once, given that:

(a) For each point  $P$ , the two endpoints of the longest line segment connecting  $P$  must be of the same colour. 

(b) For each point  $P$ , the two endpoints of the shortest line segment connecting  $P$  must be of the same colour.

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