Finding the coordinates of the corners of an aligned pole-centered spherical square

Question

Given a spherical square of radius $1$, with edge midpoints at $(1, x, 0)$, $(1, x, \pi/2$), $(1, x, \pi)$ and $(1, x,3 \pi/2)$ (in the spherical coordinate system of (radial distance, polar angle, azimuthal angle)) what are the spherical coordinates of the corners of the square?