Let \triangle A B C be an acute triangle with circumcenter O, and let Q \neq A denote the point on \odot(A B C) for which A Q \perp B C. The circumcircle of \triangle B O C intersects lines A C and A B for the second time at D and E respectively. Suppose that A Q, B C, and D E are concurrent. If O D=3 and O E=7, compute A Q.