Have b,c∈R satisfy b∈(0,1) and c>0, then let A,B denote the points of intersection of the line y=bx+c with y=∣x∣, and let O denote the origin of R2. Let f(b,c) denote the area of triangle △OAB. Let k0=20221 , and for n≥1 let kn=kn−12. If the sum ∑n=1∞f(kn,kn−1) can be written as qp for relatively prime positive integers p,q, find the remainder when p+q is divided by 1000.