Cyber Beacon Detection (Lattice Points in Disk ∩ Rectangle)

A network security administrator has identified a rogue signal originating from coordinates (xl, yl) that illuminates nodes within a radius R. The network is represented as a rectangular grid with bottom-left (x1, y1) and top-right (x2, y2). A point (x, y) is fully illuminated if its distance from the beacon's center ≤ R.

Determine how many grid nodes are completely illuminated.