124. Given a cube $A B C D A_{1} B_{1} C_{1} D_{1}$ with edge $a$. The bases of the cylinder are inscribed in the faces $A B C D$ and $A_{1} B_{1} C_{1} D_{1}$. Let $M$ be a point on the edge $A B$ such that $|A M|=a / 3$, and $N$ be a point on the edge $B_{1} C_{1}$ such that $\left|N C_{1}\right|=a / 4$. A plane passes through points $C_{1}$ and $M$, tangent to the base of the cylinder inscribed in $A B C D$, and through points $A$ and $N$ - a plane tangent to the base inscribed in $A_{1} B_{1} C_{1} D_{1}$.

Find the volume of the part of the cylinder enclosed between the planes.
124. Prove that a plane intersecting the lateral surface of a cylinder divides its volume in the same ratio as it divides the axis of the cylinder.

Answer: $\frac{\pi a^{3}}{24}$.

*) Here and below (1) denotes: Sharygin I. F. Problems in Geometry: Planimetry. - M.: Nauka, 1982 (Library "Kvant", issue 17).

125 Each lateral face of the prism is a parallelogram. If we connect the point of tangency of this face and the inscribed sphere with all the vertices of this parallelogram, then this face is divided into four triangles, and the sum of the areas of two of them, adjacent to the sides of the base, is equal to the sum of the areas of the other two. The areas of the triangles of the first type for all lateral faces will sum up to $2 S$. Therefore, the lateral surface is $4 S$, and the entire surface of the prism is $6 S$.