2002 AIME I Problem 12

Let F(z)=\frac{z+i}{z-i} for all complex numbers z\neq i, and let z_n=F(z_{n-1}) for all positive integers n. Given that z_0=\frac 1{137}+i and z_{2002}=a+bi, where a and b are real numbers, find a+b.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.