A particle of mass \(m\) and positive charge \(q\) is released from rest near the center of a cyclotron. The cyclotron consists of two hollow D-shaped conductors (dees) of outer radius \(R\) situated in a uniform magnetic field of magnitude \(B\) directed perpendicular to the plane of the dees. An alternating electric potential difference of magnitude \(\Delta V\) accelerates the particle each time it crosses the narrow gap between the dees. Assuming the particle exits the cyclotron when its circular trajectory reaches radius \(R\), which of the following expressions correctly represents the minimum number of gap crossings \(N\) required for the particle to reach the exit?