
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 disc with a radius of \( r \) is placed on a \( 10 \times 10 \) checkerboard, where each square has side length \( s \). The center of the disc is positioned at the midpoint of the side of the middle square of the checkerboard. Calculate the number of squares completely covered by the disc.

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