When rolling a certain unfair six-sided die with faces numbered 1, 2, 3, 4, 5, and 6, the probability of obtaining face \(F\) is greater than 1/6, the probability of obtaining the face opposite is less than 1/6, the probability of obtaining any one of the other four faces is 1/6, and the sum of the numbers on opposite faces is 7. When two such dice are rolled, the probability of obtaining a sum of 7 is 47/288. Given that the probability of obtaining face \(F\) is \(m/n,\) where \(m\) and \(n\) are relatively prime positive integers, find \(m+n.\)

(第二十四届AIME2 2006 第5题)