Twenty seven unit cubes are painted orange on a set of four faces so that two non-painted faces share an edge. The 27 cubes are randomly arranged to form a \(3\times 3 \times 3\) cube. Given the probability of the entire surface area of the larger cube is orange is \(\frac{p^a}{q^br^c},\) where \(p,q,\) and \(r\) are distinct primes and \(a,b,\) and \(c\) are positive integers, find \(a+b+c+p+q+r.\)
(第二十三届AIME1 2005 第9题)