What are letters doing with my numbers?

Consider a three digit number where:

\({ A }^{ 3 }+{ B }^{ 3 }+{ C }^{ 3 }+ABC\quad =\quad \overline { ABC } \)

\(\frac { A+2B+4C }{ 3 } =A\)

What is \(\overline {ABC}\) ?

Details and Assumptions: \(\overline {ABC}\) is a three digit number, whereas ABC indicates multiplication. \(\overline {ABC}\) is positive.
NOTE: A, B, and C are the digits of the number \(\overline {ABC}\). By that definition, A, B, and C are integers from 0 to 9.

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