
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

If all the black squares are removed from a standard $8 \times 8$ chessboard, the board loses the ability to accommodate even a single $2 \times 1$ domino piece. However, this ability is restored if at least one black square is returned. A chessboard reduced in this manner is referred to as an "elegantly destroyed" board.

It is not necessary to remove all black (or all white) squares for such destruction. It is possible to remove a mix of black and white squares. Thirty-two domino pieces can obviously cover the chessboard, so if at least one square is not removed in any of the formed 32 pairs of squares, then at least one domino piece can be placed on the remaining board.

To achieve a board destroyed in the required manner, at least 32 squares must be removed.

On the other hand, it is possible to remove more than 32 squares and still get an elegantly destroyed board. The trick lies in positioning these squares so that returning any one removed square allows the placement of at least one domino piece on the board. Under this strict constraint, determine the maximum number of squares that can be removed.

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