A tromino is an L-shaped tile consisting of three unit squares arranged as shown in the figure below.

Consider a \(2013 \times 2013\) chessboard whose cells are colored black and white alternatively, with the four corners colored black. Find the last three digits of the minimum number of non-overlapping trominoes needed to cover all the black cells.

**Details and assumptions**

Two trominoes are called non-overlapping if they share no common cell.

Here's an example of a \(3 \times 3\) chessboard with its cells colored black and white alternatively.

There's no restriction on whether a tromino can be placed on a white cell.

If the last three digits are \(_{\cdots }012,\) enter 12 as your answer.

This problem is not original.

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