
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  $30\times30$  table is given. We want to color some of it's unit squares such that any colored square has at most  $k$  neighbors. ( Two squares  $(i,j)$  and  $(x,y)$  are called neighbors if  $i-x,j-y\equiv0,-1,1 \pmod {30}$  and  $(i,j)\neq(x,y)$ . Therefore, each square has exactly  $8$  neighbors) 
What is the maximum possible number of colored squares if $:$  $a) k=6$  $b)k=1$ 

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