If a line $y = mx + c$ is a tangent to the circle $(x - 3)^{2} + y^{2} = 1$, and it is perpendicular to a line $L_{1}$, where $L_{1}$ is the tangent to the circle $x^{2} + y^{2} = 1$ at the point $\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$, then\\
(1) $c^{2} - 7c + 6 = 0$\\
(2) $c^{2} + 7c + 6 = 0$\\
(3) $c^{2} + 6c + 7 = 0$\\
(4) $c^{2} - 6c + 7 = 0$