Let $r$, $s$, and $t$ be the roots of the polynomial $x^3 + 6x^2 + 7x + 8$. Calculate the expression:
\[
(r^2 + s + t)(s^2 + t + r)(t^2 + r + s)
\]

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