Two thin circular discs, one with a mass $m$ and radius $a,$ and the other with a mass $4m$ and radius $2a,$ are rigidly fixed by a massless, rigid rod of length $l=\sqrt[\phantom{2}]{2}$a through their centers. This assembly is laid on a firm, flat surface. The discs are set rolling without slipping on the surface, such that the entire assembly rotates about the axis of the rod with an angular speed $.$

Consider the following additional details:

The rod is horizontal and lies along the x$-$axis, with the center of the smaller dis at the origin.

The center of the larger disc is located at $x=\sqrt[\phantom{2}]{2}a.$

The z$-$axis is perpendicular to the plane of the discs and the flat surface.

The assembly rolls in the $x-$y plane without slipping, meaning that the velocity o the point of contact with the ground is zero at any instant.

The assembly's angular momentum about the point $O($the origin$)$ is denoted as vec$(L).$ We need to analyze the motion and angular momentum properties of this complex rolling system. Determine which of the following statements are true:

$($A$)$ The center of mass of the assembly rotates about the $z-$axis with an angular speed of $\frac{}{5}.$

$($B$)$ The magnitude of the angular momentum of the center of mass of the assembly about the point $O$ is $81m{a}^2.$

$($C$)$ The magnitude of the angular momentum of the assembly about its center of mass is $\frac{17}{2}m{a}^2.$

$($D$)$ The magnitude of the z$-$component of vec$(L)$ is $55m{a}^2.$