
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 a positive integer  $n\geq 2$ . Let  $A=\{ (a,b)\mid a,b\in \{ 1,2,…,n\} \}$  be the set of points in Cartesian coordinate plane. How many ways to colour points in  $A$ , each by one of three fixed colour, such that, for any  $a,b\in \{ 1,2,…,n-1\}$ , if  $(a,b)$  and  $(a+1,b)$  have same colour, then  $(a,b+1)$  and  $(a+1,b+1)$  also have same colour.

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