FKMO 2008 Problem 3

Determine all functions $f : \mathbb{R}^+\rightarrow\mathbb{R}$ that satisfy the following $f(1)=2008$ , $|{f(x)}| \le x^2+1004^2$ , $f\left (x+y+\frac{1}{x}+\frac{1}{y}\right )=f\left (x+\frac{1}{y}\right )+f\left (y+\frac{1}{x}\right ).$