Let $a, b, c, d, e,$ and $f$ be real numbers. Define the polynomials: 
\[ P(x) = 2x^4 - 26x^3 + ax^2 + bx + c \] 
and 
\[ Q(x) = 5x^4 - 80x^3 + dx^2 + ex + f. \] 
Let $S$ be the set of all complex numbers which are a root of either $P$ or $Q$ (or both). Given that $S = \{1,2,3,4,5\}$, compute $P(6) \cdot Q(6).$

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