Use the Gauss$-$Jordan reduction to solve the following linear system:

$\{\begin{array}{c}{x}_1-x_2+4x_3=5\end{array}$

$5x_1-6x_2+6x_3=4$

$3x_1+54x_3=78$

$\left[\begin{array}{c}{x}_1\end{array}\right]$

$x_2$

$x_3=[]+s[]$