Square A B C D has center O, A B=900, E and F are on \overline{A B} with A E<B F and E between A and F, m \angle E O F=45^{\circ}, and E F=400. Given that B F=p+q \sqrt{r}, where p, q, and r are positive integers and r is not divisible by the square of any prime, find p+q+r.