Two rigid glass bulbs, Bulb 1 of volume \(V_0\) and Bulb 2 of volume \(2V_0\), are connected by a narrow tube of negligible volume equipped with a closed stopcock. Bulb 1 is held at a constant absolute temperature \(T_0\) by a cold thermal bath, while Bulb 2 is held at a constant absolute temperature \(2T_0\) by a hot thermal bath. Initially, Bulb 1 contains an ideal gas at pressure \(P_0\) and Bulb 2 is completely evacuated. The stopcock is opened, allowing gas to flow between the bulbs until thermal and mechanical equilibrium is established while the baths maintain the temperatures of the respective bulbs. What is the final equilibrium pressure \(P_f\) of the gas system in terms of \(P_0\)?