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There may be a few ways to do that. This is how I would do it.
1 - Duplicate the circle, and leave it on top of the rectangles (if you use Cut Path with the rectangles on top, it will cut the circle, and not cut the rectangles)
2 - Select the circle and the yellow rectangle
3 - Path menu > Cut Path (it might look like nothing happened, but in fact, the lines of the rectangle were cut where they intersect with the circle)
4 - Select and delete the unwanted parts of the rectangles
5 - Repeat #1 through #4 for the red rectangle.
Edit
Oops, I meant to ask or comment about the issue you mentioned about 100 lines or 1000 words. Does that refer to the gcode? Sometimes people have trouble with making files for cutting, where there are hundreds of lines or paths in the file, which need to be combined. I don't see that in your file, but I just wanted to make sure.