Let $\triangle ABC$ be a triangle such that the area $[ABC] = 10$ and $\tan(\angle ABC) = 5$. If the smallest possible value of $(\overline{AC})^2$ can be expressed as $-a + b\sqrt{c}$ for positive integers $a, b, c$, what is $a + b + c$?

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