Let's count!

A domino is a \(2\times1\) or \(1\times2\) tile.

Let \(N\) be the number of ways placing \(100\) dominoes without overlapping on a \(20\times20\) chessboard so that every \(2\times2\) square contains at least two uncovered unit squares which lie in the same row or column.

Find the last 4 digits of \(N\).

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