Dutch Mathematical Olympiad

In a square, we draw the inscribed circle, into which we draw an inscribed square whose sides are parallel to the sides of the original square. We colour the area between these squares dark grey. In the smaller square, we again draw the inscribed circle containing another inscribed square with parallel sides; the area between these we colour light grey. We repeat this process until there are 2023 dark grey and 2023 light grey areas. In the picture, you can see the first steps drawn. The inner square remains white. The area of all the dark grey areas together is exactly 1. What is the area of all the light grey areas together?