Japan Mathematical Olympiad

A square $ABCD$ is given. Points $P$, $Q$, $R$, $S$ lie on the sides $AB$, $BC$, $CD$, $DA$, of this square, respectively, in such a way that the lines $PR$ and $BC$ are parallel and so are the lines $SQ$ and $AB$. Let $Z$ be the point of intersection of the lines $PR$ and $SQ$. If $BP=7$, $BQ=6$, $DZ=5$, determine the length of a side of the square $ABCD$. Here we are denoting the length of a line segment $XY$ also by $XY$.