One Parameter, Three Fates

Let $k$ be a real parameter and consider the system

$\begin{aligned}x + 2y + z &= 3\\ 2x + ky + 3z &= 5\\ 3x + 6y + kz &= 9\end{aligned}$

whose coefficient matrix and right-hand side are

$A = \begin{pmatrix}1 & 2 & 1\\ 2 & k & 3\\ 3 & 6 & k\end{pmatrix}, \qquad b = \begin{pmatrix}3\\ 5\\ 9\end{pmatrix}$

Classify the system completely: for which values of $k$ does it have no solution, for which does it have infinitely many, and for which does it have exactly one?