Strange Radical

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:

\[\Large \alpha+\beta+\gamma=\ ?\]

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