.
I used the online free of charge matrix solver
https://matrix.reshish.com/gaussSolution.php
which, in particular, makes the Gauss-Jordan elimination.
So, I inputted the augmented matrix and pressed the "Solve" button.
The solver does not select a most efficient way, but, in any case, makes his work from the begginning to the very end.
Below see the results:
Your matrix
X1 X2 X3 b
1 0 -2 3 1
2 3 6 -3 -2
3 6 6 3 5
Make the pivot in the 1st column by dividing the 2nd row by 3 and swap the 2nd and the 1st rows
X1 X2 X3 b
1 1 2 -1 -2/3
2 0 -2 3 1
3 6 6 3 5
Multiply the 1st row by 6
X1 X2 X3 b
1 6 12 -6 -4
2 0 -2 3 1
3 6 6 3 5
Subtract the 1st row from the 3rd row and restore it
X1 X2 X3 b
1 1 2 -1 -2/3
2 0 -2 3 1
3 0 -6 9 9
Make the pivot in the 2nd column by dividing the 2nd row by -2
X1 X2 X3 b
1 1 2 -1 -2/3
2 0 1 -3/2 -1/2
3 0 -6 9 9
Multiply the 2nd row by 2
X1 X2 X3 b
1 1 2 -1 -2/3
2 0 2 -3 -1
3 0 -6 9 9
Subtract the 2nd row from the 1st row and restore it
X1 X2 X3 b
1 1 0 2 1/3
2 0 1 -3/2 -1/2
3 0 -6 9 9
Multiply the 2nd row by -6
X1 X2 X3 b
1 1 0 2 1/3
2 0 -6 9 3
3 0 -6 9 9
Subtract the 2nd row from the 3rd row and restore it
X1 X2 X3 b
1 1 0 2 1/3
2 0 1 -3/2 -1/2
3 0 0 0 6
Do you see this last line of the transformed matrix which consists of zero coefficients with non-zero right side ?
It means that the system is INCONSISTENT.