Consider an \(N \times N\) grid, initially completely unpopulated. We can mark squares in this grid in at least \(2^{(N \cdot N)}\) ways, such as for a \(2 \times 2\) grid:

Of these \(16\) grids, precisely \(13\) are *continuous*. That is, there exists a path, using edge adjacency (no corners/diagonals) from every black square to every other black square, and there is at least one black square.

Of all \(3 \times 3\) grids, \(218\) are continuous.

How many \(10 \times 10\) grids are continuous?

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