A block of mass \(m_0\) slides along a frictionless horizontal surface with initial kinetic energy \(K_0\). It undergoes a completely inelastic collision with a mass-accumulating target initially at rest. During the collision process, which lasts from time \(t = 0\) to \(t = T\), the total mass of the combined object increases according to the function \(m(t) = m_0 e^{kt}\), where \(k\) is a positive constant. Assuming no external horizontal forces act on the system, what is the final kinetic energy of the combined mass at time \(t = T\)?