Tesla embraces the heat engine

Tesla decides to go back to basics with their new line, the Entropica series. Leaving aside their rather successful venture into electric engines, Tesla pivots to embrace the heat engine, which uses two heat blocks of initial temperature $T_+$, and $T_-$ to drive an engine through a thermodynamic cycle (pictured above).

Concretely, each cycle drives some heat $\delta Q$ from the hot reservoir to the cold reservoir, and the engine performs some work $\delta W$ to propel the car, such that the engine returns to its original state after each cycle. Clearly, each cycle lowers the temperature $T_+$, and raises the temperature $T_-$, until $T_+=T_-=T^*$, at which point the engine can do no further work.

Find $T^*$ (in deg Kelvin) at the point where the engine stops running.

Assumptions and Details

• The entire process is thermodynamically reversible.
• $T_+$ = 423$^\circ$ K
• $T_-$ = 300$^\circ$ K
• Both heat reservoirs are metal blocks of heat capacity $\gamma$