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 $a+b+c+p+q+r.$



Let's think step by step and output the final answer within \boxed{}.
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