Consider a \(10 \times 11\) rectangular room which is divided into \(110\) unit squares in the natural way by lines parallel to the sides of the room. We also have a collection of strange tiles. Each tile consists of six unit squares connected together in the following shape:
What is the maximal possible number of non-overlapping tiles we can pack into the room such that each tile is covering exactly six unit squares of the room? Give proof and diagram of the tiling.