10. (20 points) Given the sequence $\left\{x_{n}\right\}$ satisfies
$$
x_{1}=\frac{1}{3}, x_{n+1}=x_{n}^{2}+x_{n}(n=1,2, \cdots) \text {. }
$$

Find the value of $\sum_{n=1}^{2002} \frac{1}{x_{n}+1}$ between which two integers.