Find the density of states in X points of Silicon

The problem statement is given verbatim

In Si, the dispersion relation at the [001] X points is: $$E=frac{hbar^2}{2}left(frac{k_x^2+k_y^2}{m_t}+frac{(k_z-G)^2}{m_l}right)$$ where G is the reciprocal lattice vector at X points. Find the total density of states for the X points in silicon.

So my question is: how do we solve this problem (step by step)?

To clarify, I believe the X points are the X symmetry points for a fcc lattice. The $m_l$ and $m_t$ are the longitudinal and the transverse effective masses for silicon. I personally think $G=frac{2pi}{a}$ for the first Brillouin zone, where a is the conventional simple cubic side length so that $G$ equals the distance between the $Gamma$ point and the $X$ point. I am just confused on the purpose of this problem. Since the X symmetry points are discrete points in the reciprocal space, it seems weird that we are supposed to find density of states for these discrete points.

My own feeling is that there are multiple such (discrete) X-points with different $k_x$, $k_y$, and $k_z$, and they may correspond to different energies in the conduction band. In the first Brillouin zone, for each conduction band, there are six X-points, corresponding to the same energy. Then basically we are to locate each set of 6 X-points in each conduction band in the first Brillouin zone, and then we will use the Dirac delta function to write out the density of states peaked at these X-point energies. Then we will want to extend the result to all the Brillouin zones, which seems rather strange to me. And it seems overly difficult to carry out the computation for the scope of this problem’s context.

I am not sure if I am understanding this problem properly, or the problem might just not be well-phrased.