Let $a$, $b$, and $c$ be real numbers satisfying the equations:
\[
a^3 + abc = 26 \\
b^3 + abc = 78 \\
c^3 - abc = 104
\]
Find $a^3 + b^3 + c^3$.

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