Block A of mass \(m\) and block B of mass \(2m\) are on a horizontal, frictionless surface. Block A moves to the right with speed \(4v_0\), and block B moves to the left with speed \(v_0\), as shown. A light spring is attached to the front of block B. The blocks collide, compressing the spring until the blocks reach a common velocity at the moment of maximum compression.

Let \(J\) be the magnitude of the impulse delivered to block A from the start of the collision until the moment of maximum compression, and let \(P_{sys}\) be the magnitude of the total momentum of the two-block system. What is the ratio \(J/P_{sys}\)?