Decomposing a matrix that has duplicate columns using PA=LU factorization

Question

I am given the following matrix.

$$ A = \begin{bmatrix} 3 & 3 & 9 & 6 \\ 4 & 4 &4 &4 \\ 1 & 1 & 5 & 5 \\ 2 & 2 & 4 & 6\end{bmatrix} $$

As you can notice, the first two columns are identical. Hence, after pivoting (getting $P_1$) and then finding the first Gauss transform $L_1$, the matrix $P_1 L_1 A$ becomes

$$ P_1 L_1 A = \begin{bmatrix} 4 & 4& 4&4\\0&0&6&3 \\0&0&4&4 \\0&0&2&4 \end{bmatrix} $$

1. What should be the next step after this?
   
   Because I'm used to working with this: If the first pivot in the first column, then after finding the $P_1$ and $L_1$, I expect to find the next pivot in the second column and then finding $P_2$ and $L_2$ and so on and so forth. But this matrix is a different scenario. What am I supposed to do after finding the $P_1$ and $L_1$ and then find out that the next pivot is in the third column instead of the second column?

2. If this is a problem from a specific textbook, does anyone know what that textbook is?