A battery with an electromotive force (emf) \(\varepsilon\) and internal resistance \(r\) is connected to an external circuit. The external circuit consists of a single resistor of resistance \(R\) connected in series with a parallel combination of two identical resistors, each also having resistance \(R\). In terms of \(\varepsilon\), \(R\), and \(r\), what is the rate at which energy is dissipated as thermal energy in the battery's internal resistance?