Minimum Cost of Two Access Roads

Town $B$ is $4$ km due east of town $A$. Town $C$ is $2$ km from $B$ in the direction N$30^\circ$E. Along the riverbank, modelled by the curve $PQ$, every point is exactly $2$ km farther from $A$ than from $B$. A dock $M$ is to be built at some point on the curve $PQ$ to transport goods to towns $B$ and $C$. Building the road from $M$ to $B$ costs $a$ (in ten-thousand yuan) per km, and building the road from $M$ to $C$ costs $2a$ (in ten-thousand yuan) per km. Find the minimum total cost of building these two roads.