Let $$F(z)=\frac{z+i}{z-i}$$ for all complex numbers $$z\not= i$$, 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 $$a+b$$.

(第二十届AIME1 2002 第12题)