Let ω be a circle centered at O with radius R=2018. For any 0<r<1009, let γ be a circle of radius r centered at a point I satisfying OI=R(R−2r). Choose any A,B,C∈ω with AC,AB tangent to γ at E,F, respectively. Suppose a circle of radius rA is tangent to AB,AC, and internally tangent to ω at a point D with rA=5r. Let line EF meet ω at P1,Q1. Suppose P2,P3,Q2,Q3 lie on ω such that P1P2,P1P3,Q1Q2,Q1Q3 are tangent to γ. Let P2P3,Q2Q3 meet at K, and suppose KI meets AD at a point X. Then as r varies from 0 to 1009, the maximum possible value of OX can be expressed in the form cab, where a,b,c are positive integers such that b is not divisible by the square of any prime and gcd(a,c)=1. Compute 10a+b+c.Proposed by Vincent Huang