Intersecting planes, distances and a family of parallel planes

Find the point of intersection of the three planes.

Find a vector that gives the direction of the line of intersection of the planes $\Pi_2$ and $\Pi_3$.

Find a set of parametric equations that give the coordinates of the points that lie on this line of intersection.

Find the shortest distance from the point $(1, 1, 2)$ to the plane $\Pi_3$.

Consider a family of planes, $\Psi$, with equation $6x + 27z = m$, where $m \in N$.

Show that the plane $\Pi_1$ is parallel to each member of $\Psi$.

Find all values of $m$ for which the shortest distance between plane $\Pi_1$ and the plane of the form $6x + 27z = m$ is $\frac{23}{3\sqrt{85}}$.