There is a regular tetrahedron OABC with edge length 6. Let $S _ { 1 } , S _ { 2 } , S _ { 3 }$ be the orthogonal projections onto plane ABC of the three circles inscribed in triangles $\triangle \mathrm { OAB } , \triangle \mathrm { OBC } , \triangle \mathrm { OCA }$ respectively.\\
As shown in the figure, let $S$ be the area of the dark region enclosed by the three figures $S _ { 1 } , S _ { 2 } , S _ { 3 }$. Find the value of $( S + \pi ) ^ { 2 }$. [4 points]