Holiday Contest 2025 - Guts Round - Set 5 Problem 1

A a,b-lattice grid is defined by the set of all points (ax+b,ay) such that x, y, \in \mathbb{Z}. Consider the 5,2-lattice grid and the 3,0-lattice grid. The probability that a random point picked on the coordinate plane (not necessarily a lattice point) is less than 1 unit away from some point in the 5,2-lattice grid and the 3,0-lattice grid can be expressed as \frac{p\pi-q}{r} where p, q, and r are positive integers with greatest common divisor 1. Compute p+q+r.