MD-MO 2007 Problem 1

Define the sequence $(x_{n})$ : $x_{1}=\frac{1}{3}$ and $x_{n+1}=x_{n}^{2}+x_{n}$ . Find $\left[\frac{1}{x_{1}+1}+\frac{1}{x_{2}+1}+\dots+\frac{1}{x_{2007}+1}\right]$ , wehere $[$ $]$ denotes the integer part.