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Two points. These may coincide if the line is a tangent to the sphere.
The answer will depend on how the triangle is situated within the rectangle (how many of the triangle's vertices coincide with those of the rectangle), and what other information you have.
If the lines intersect, then the intersection point is the solution of the system. If the lines coincide, then there are infinite number of the solutions for the system. If the lines are parallel, there is no solution for the system.