Calculus Level 5

$\Large\mathfrak T(\color{blue}{a})=\sqrt{\dfrac{\color{blue}{a}^{F_1}}{\sqrt{\dfrac{\color{blue}{a}^{F_2}}{\sqrt{\dfrac{\color{blue}{a}^{F_3}}{\sqrt{\cdots}}}}}}}~~,\color{blue}{a}>0$

where $$F_n$$ denotes the $$n^\text{th}$$ Fibonacci number.

Then $$\large\mathfrak T(\color{blue}{4})=\alpha^{\frac{\beta}{\gamma}}$$ ,where $$\alpha$$ is a prime and $$\beta$$ and $$\gamma$$ are co prime positive integers. Then:

$\Large \alpha+\beta+\gamma=\ ?$

×