Solve for **integers** a, b, c:

\[ a+b=1-c \] \[ a^{3}+b^{3}=1-c^{2} \]

If the solutions you obtain for example, (1,2,3) and (4,5,6) then your answer will be \( 1+2+3+4+5+6 = \boxed{21}\).

Disregard the solution family: \( (m, -m, 1) \).

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