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)