
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
Let  $n$  be positive integer and  $S$ = { $0,1,…,n$ }, Define set of point in the plane.   $$ A = \{(x,y) \in S \times S \mid -1 \leq x-y \leq 1 \}  $$  , We want to place a electricity post on a point in  $A$  such that each electricity post can shine in radius 1.01 unit. Define minimum number of electricity post such that every point in  $A$  is in shine area

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