Parallel lines, a parallelogram and a midplane

The points $A,B$ have position vectors $3\mathbf i+\mathbf j$ and $3\mathbf i+3\mathbf j$ respectively. The line $l_1$ and the planes $\Pi_1,\Pi_2$ have equations as follows:

$l_1:\mathbf r=\overrightarrow{OA}+\alpha(2,1,-1)$, $\Pi_1:x+2z=3$, $\Pi_2:\mathbf r=\lambda(1,1,0)+\mu(0,1,1)$, where $\alpha,\lambda,\mu\in\mathbb R$.

It is given that the planes $\Pi_1,\Pi_2$ intersect in the line $l_2$.

(i) Find a vector equation of the line $l_2$ and show that the line $l_2$ is parallel to the line $l_1$. Hence find the shortest distance between the lines $l_1,l_2$.

(ii) The points $C,D$ are on the lines $l_1,l_2$ respectively such that $\angle ABC=90^\circ$ and $ABDC$ is a parallelogram. Find the position vectors of $C,D$. Hence find the area of the parallelogram $ABDC$.

(iii) The plane $\Pi_3$ is parallel to the plane $\Pi_2$ and is equidistant to both point $A$ and the plane $\Pi_2$. Show that the equation of the plane $\Pi_3$ is given by $\mathbf r\cdot(\mathbf i-\mathbf j+\mathbf k)=1$.

(iv) Find the position vector of the foot of the perpendicular from the point $A$ to the plane $\Pi_3$.

Source wording note: In the first sentence of part (iii), the question PDF prints $\Pi_2$ where the subsequent reference to $\Pi_3$ identifies the intended plane. This transcription uses $\Pi_3$ there; the original wording is recorded as a source typo.