As shown in the figure, a cylinder with base radius 7 and a cone with base radius 5 and height 12 are placed on a plane $\alpha$, and the circumference of the base of the cone is inscribed in the circumference of the base of the cylinder. Let O be the center of the base of the cylinder that meets plane $\alpha$, and let A be the apex of the cone. A sphere $S$ with center B and radius 4 satisfies the following conditions. (가) The sphere $S$ is tangent to both the cylinder and the cone. (나) When $\mathrm { A } ^ { \prime }$ and $\mathrm { B } ^ { \prime }$ are the orthogonal projections of points $\mathrm { A }$ and $\mathrm { B }$ onto plane $\alpha$ respectively, $\angle \mathrm { A } ^ { \prime } \mathrm { OB } ^ { \prime } = 180 ^ { \circ }$. When the acute angle between line AB and plane $\alpha$ is $\theta$, $\tan \theta = p$. Find the value of $100 p$. (Note: The center of the base of the cone and point $\mathrm { A } ^ { \prime }$ coincide.) [4 points]
As shown in the figure, a cylinder with base radius 7 and a cone with base radius 5 and height 12 are placed on a plane $\alpha$, and the circumference of the base of the cone is inscribed in the circumference of the base of the cylinder. Let O be the center of the base of the cylinder that meets plane $\alpha$, and let A be the apex of the cone. A sphere $S$ with center B and radius 4 satisfies the following conditions.\\
(가) The sphere $S$ is tangent to both the cylinder and the cone.\\
(나) When $\mathrm { A } ^ { \prime }$ and $\mathrm { B } ^ { \prime }$ are the orthogonal projections of points $\mathrm { A }$ and $\mathrm { B }$ onto plane $\alpha$ respectively, $\angle \mathrm { A } ^ { \prime } \mathrm { OB } ^ { \prime } = 180 ^ { \circ }$.
When the acute angle between line AB and plane $\alpha$ is $\theta$, $\tan \theta = p$. Find the value of $100 p$. (Note: The center of the base of the cone and point $\mathrm { A } ^ { \prime }$ coincide.) [4 points]