$(30\backslash \%)$ Figure shows two power cycles, denoted $1$ and $2,$ operating in series, together with three thermal reservoirs. The energy transfer by heat into cycle $2$ is equal in magnitude to the energy transfer by heat from cycle $1.$ All energy transfers are positive in the direction of the arrows.

a$)$ If cycles $1$ and $2$ are each irreversible, derive an expression for the thermal efficiency of an overall cycle consisting of cycles $1$ and $2$ in terms of their individual thermal efficiencies.

b$)$ If cycles $1$ and $2$ are each reversible, use the result of part $($a$)$ to obtain an expression for the thermal efficiency of the overall cycle in terms of the temperatures of reservoirs. Comment.

c$)$ If cycles $1$ and $2$ are each reversible and have the same thermal efficiency, obtain an expression for the intermediate temperature $(\backslash ($ T $\backslash ))$ in terms of $\backslash ($ T$\_\{\backslash$mathrm$\{$H$\}\}\backslash )$ and $\backslash ($ T$\_\{\backslash$mathrm$\{$C$\}\}\backslash ).$