How to find the number of intersection points of a circle and a square?

Question

Geometry

Below is the problem with the solution of which there were difficulties.
How to find the number of intersection points?
Options 3,4,7 caused difficulties.
Given the coordinates of the center of the circle and its radius, as well as the coordinates of the center of the
square and its side length.
Find the relative position of the figures.
Sketch of options:
1) A circle is inscribed in a square
2) A circle is circumscribed around a square
3) A circle and a square touch at one point (external and internal
touch)
4) A circle and a square touch at two points
5) The circle lies inside the square
6) The square lies inside circles
7) A circle and a square intersect at two or more points

Answer 1

Write first functions to determine the relative position with a vertical and horizontal segment.
Then, knowing the answers for each side, you can already find the answer for the square.