Cross Twice or More

How many ways are there to get from the bottom left corner of a \(10\times 10\) grid to the top right corner in 20 steps, if you must cross the main diagonal (in red) two times or more?

In the above example, the blue path crosses the red diagonal three times, so it is a valid path.

The paths are monotonic, i.e, only moves to the right and the top are allowed.

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