5.3.1.8 Construction of sections

Condition: Point \( O \) and \( \mathrm{O}_{1} \)-centers of faces \( A B C D \) and \( A_{1} B_{1} C_{1} D_{1} \) of cube \( A B C D A_{1} B_{1} C_{1} D_{1}\). On the segment \( O O_{1} \) a point \( S \) is taken so that \( O_{1} S: O S=1: 3 \). Through this point a section of the cube is drawn parallel to its diagonal \( A C_{1} \) and the diagonal \( B D \) of the base. Find the cross-sectional area if the length of the edge of the cube is \( a \).