Consider the polynomial $x^3 - 3x^2 + 10$. Let $a$, $b$, and $c$ be its roots. Compute $a^2b^2c^2 + a^2b^2 + b^2c^2 + c^2a^2 + a^2 + b^2 + c^2$.

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