Consider the system of equations

x^2 + y^2 = R^2 (circle)

|x| + |y| = 1 (square)

I need to find the number of solutions to the two equations for different values of R.

I know if R>1 there are 0 solutions, 4 if R=1. But there is a value, t, of R for which the circle intersects the square 8 times i.e. t<R<1. My problem is how to find the value t?