In Tiger Electronics' handheld electronic solitaire game Lights Out, the player strives to turn out all $25$ lights that make up a $55$ grid of cells. On each turn, the player is allowed to click on any one cell. Clicking on a cell activates a switch that causes the states of the cell and its neighbors to change from on to off or from off to on$.$ Corner cells are considered to have two neighbors, edge cells to have three, and interior cells to have four. The diagram demonstrates what happens when the player clicks on cells $(1,1)$ and $(1,2).$

Formulate an integer program for finding a way to turn out all the lights in as few turns as possible. Hints: $(1)$ The order in which the cells are clicked doesn't matter. $(2)$ A cell needn't be clicked more than once.