Spring with Thermodynamics

Two heat-insulating pistons connected by spring can slide without friction inside a horizontal heat insulating tube, which is open at both of its ends. One mole of a mono-atomic gas is filled between the pistons at temperature $T_{1}.$

At this temperature of the gas, the spring is not relaxed. If an amount of heat is supplied to the gas, its temperature becomes $T_{2}$ and the length of spring becomes $\alpha$ times of its initial length.

If the heat is supplied to the gas can be expressed as $\beta R \left [ \delta ( T_{2}-T_{1} ) + \frac{ \alpha+ \gamma }{ \beta \alpha } \cdot ( \alpha T_{1} +T_{2} )\right ],$
submit your answer as the sum of constants $\beta + \gamma +\delta .$