Let

$$A=\left(\begin{array}{cccc} 1 & a & a^{2} & a^{3} \\ a^{3} & 1 & a & a^{2} \\ a^{2} & a^{3} & 1 & a \\ a & a^{2} & a^{3} & 1 \end{array}\right), \quad b=\left(\begin{array}{c} \gamma \\ 0 \\ 0 \\ \gamma a \end{array}\right), \quad \gamma=1-a^{4} \neq 0$$

Find the LU factorization of the matrix $A$ and use it to solve the system $A x=b$.