Squares In Between

\[\begin{cases} a + b + c = 11\\ a^3 + b^3 + c^3 = 785 \end{cases}\]

If \(a, b, c\) are the integers, whose sum of any two variables is larger than 1, that satisfy the system above, what is the value of \(a^2 + b^2 + c^2\)?

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