Proof Writing & Problem Solving · Superchallenge problem

Each cell of a $7 \times 7$ table holds $+1$ or $-1$. Multiplying the seven entries of each row and of each column gives $14$ products, each equal to $+1$ or $-1$. Let $S$ be the sum of these $14$ products. How many different values can $S$ take, over all such tables?