
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
What is the minimum number of pieces required to place on the squares of a chessboard of size

a) $8 \times 8$ cells,

b) $n \times n$ cells

such that on each line passing through the center of any arbitrary square and parallel to any side or diagonal of the board, there is at least one piece? (Pieces are placed in the centers of the squares.)

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