Use Gaussian elimination to find the complete solution to the following system of equations, or show that none exists. \[ \left\{\begin{array}{rr} 5 x+12 y+5 z= & 13 \\ 2 x+5 y+5 z= & -8 \\ x+2 y-5 z= & 3 \end{array}\right. \] Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. There is one solution. The solution set is $\{(,, D)\}$. (Simplify your answers.) B. There are infinitely many solutions. The solution set is $\{(, ;, z)\}$, where $z$ is any real number. (Type expressions using $z$ as the variable. Use integers or fractions for any numbers in the expression.) C. There is no solution. The solution set is $\varnothing$.

Step 1 ：We start by writing the system of equations in augmented matrix form: \[\begin{bmatrix} 5 & 12 & 5 & 13 \\ 2 & 5 & 5 & -8 \\ 1 & 2 & -5 & 3 \end{bmatrix}\]

Step 2 ：We then perform row operations to get the matrix in row-echelon form. First, we swap the first and third rows to get: \[\begin{bmatrix} 1 & 2 & -5 & 3 \\ 2 & 5 & 5 & -8 \\ 5 & 12 & 5 & 13 \end{bmatrix}\]

Step 3 ：Next, we subtract twice the first row from the second row and five times the first row from the third row to get: \[\begin{bmatrix} 1 & 2 & -5 & 3 \\ 0 & 1 & 15 & -14 \\ 0 & 2 & 30 & -2 \end{bmatrix}\]

Step 4 ：We then subtract twice the second row from the third row to get: \[\begin{bmatrix} 1 & 2 & -5 & 3 \\ 0 & 1 & 15 & -14 \\ 0 & 0 & 0 & 30 \end{bmatrix}\]

Step 5 ：From the third row, we see that 0 = 30, which is a contradiction. Therefore, the system of equations has no solution.

Step 6 ：So, the correct choice is C. There is no solution. The solution set is \(\varnothing\).