
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
Place pieces on a square grid chessboard such that every line passing through the center of any square and parallel to one of the sides or one of the diagonals of the chessboard has at least one piece (the pieces are placed at the center of each square). Find the minimum number of pieces required for:
(1) an $8 \times 8$ chessboard;
(2) an $n \times n$ chessboard.

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