
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
Consider the points  $P$  = $(0,0)$ , $Q$  =  $(1,0)$ ,  $R$ =   $(2,0)$ ,  $S$  = $(3,0)$  in the  $xy$ -plane. Let  $A,B,C,D$  be four finite sets of colours(not necessarily distinct nor disjoint). In how many ways can  $P,Q,R$  be coloured bu colours in  $A,B,C$  respectively if adjacent points have to get different colours? In how many ways can  $P,Q,R,S$  be coloured by colours in  $A,B,C,D$  respectively if adjacent points have to get different colors?

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