Let the image of the point \( P(1,6,a) \) in the line \( L : \, \dfrac{x}{1} = \dfrac{y-1}{2} = \dfrac{z-a+1}{b}, \, b>0 \), be \( \left(\dfrac{a}{3},0,a+c\right) \). If \( S(\alpha,\beta,\gamma), \, \alpha>0 \), is the point on \( L \) such that the distance of \( S \) from the foot of perpendicular from the point \( P \) on \( L \) is \( 2\sqrt{14} \), then \( \alpha + \beta + \gamma \) is equal to:
