The terms of the sequence \left(a_{i}\right) defined by a_{n+2}=\frac{a_{n}+2009}{1+a_{n+1}} for n \geq 1 are positive integers. Find the minimum possible value of a_{1}+a_{2}.
The terms of the sequence \left(a_{i}\right) defined by a_{n+2}=\frac{a_{n}+2009}{1+a_{n+1}} for n \geq 1 are positive integers. Find the minimum possible value of a_{1}+a_{2}.