Let f_{1}(x)=\frac{2}{3}-\frac{3}{3 x+1}, and for n \geq 2, define f_{n}(x)=f_{1}\left(f_{n-1}(x)\right). The value of x that satisfies f_{1001}(x)=x-3 can be expressed in the form \frac{m}{n}, where m and n are relatively prime positive integers. Find m+n.