A cart of mass \(m_1\) travels to the right with initial speed \(v_0\) along a frictionless horizontal track toward a stationary cart of mass \(m_2\). The two carts collide and stick together, then immediately enter a frictionless vertical circular loop of radius \(R\). In terms of \(m_1\), \(m_2\), \(R\), and fundamental constants, what is the minimum initial speed \(v_0\) required for the combined carts to complete the loop without losing contact with the track?