
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
There is an infinite grid plane where no point is colored blue. For one ruble, you can choose a cell and paint all its sides blue, even if one of its sides was already painted. What is the minimum amount needed to obtain a grid square of $1001 \times 1001$, with all grid lines inside and on the border painted blue? It is not required that ultimately only the grid lines of the square are painted.

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