A circuit consists of an ideal battery of emf \(\varepsilon\), an open switch \(S\), three resistors with resistances \(R_1 = R\), \(R_2 = 2R\), and \(R_3 = R\), and an initially uncharged capacitor of capacitance \(C\). Branch 1 contains resistors \(R_1\) and \(R_2\) in series with node \(X\) located between them. Branch 2 contains capacitor \(C\) and resistor \(R_3\) in series with node \(Y\) located between them. At time \(t = 0\), switch \(S\) is closed. What are the potential difference \(V_X - V_Y\) immediately after the switch is closed (\(t = 0^+\)) and the initial rate of change of this potential difference, \(\left.\dfrac{d(V_X - V_Y)}{dt}\right|_{t=0^+}\)?