Problem 3. (4 points) On a table, a cardboard circle with a radius of 5 cm is placed. As long as possible, Peter places cardboard squares with a side of 5 cm outside the circle in such a way that the following conditions are met:

1) one vertex of each square lies on the boundary of the circle;
2) the squares do not overlap;
3) each new square shares a vertex with the one placed before it.

![](https://cdn.mathpix.com/cropped/2024_06_03_336fa2e3a24e7097f2a8g-1.jpg?height=363&width=333&top_left_y=1686&top_left_x=1465)

Find the number of squares Peter can place and prove that the first and the last of them also share a vertex.