If the distances of the point \( (1,2,a) \) from the line \( \dfrac{x-1}{1} = \dfrac{y}{2} = \dfrac{z-1}{1} \) along the lines \( L_{1}: \ \dfrac{x-1}{3} = \dfrac{y-2}{4} = \dfrac{z-a}{b} \) and \( L_{2}: \ \dfrac{x-1}{1} = \dfrac{y-2}{4} = \dfrac{z-a}{c} \) are equal, then \( a+b+c \) is equal to:
