12.1.23 Olympic geometry

Condition: At the base of the quadrangular pyramid \( S A B C D \) lies a square \( A B C D \) with side \( A B=9 \). On the continuation of the diagonal \( C A \) for the point \( A \) the point \( H \) is chosen so that \( A H=4 C A \). The segment \( S H=5 \) is perpendicular to the plane of the base of the pyramid. What is the largest volume \(V\) that a cylinder can have if it is located inside a pyramid so that one of its bases lies on the base of the pyramid? In your answer, indicate the value \(