The equation $$2^{333x-2} + 2^{111x+2} = 2^{222x+1} + 1$$ has three real roots. Given that their sum is $$\frac mn$$ where $$m$$ and $$n$$ are relatively prime positive integers, find $$m+n.$$

(第二十三届AIME1 2005 第8题)