Okay, That's A Lot Of Integers

Number Theory Level 5

\[ x+y+z=3 \] \[ x^3+y^3+z^3=3 \] Let \( \displaystyle a \) be the sum of all possible integer values of \( \displaystyle x \).

Let \( \displaystyle b \) be the sum of all possible integer values of \( \displaystyle y \).

Let \( \displaystyle c \) be the sum of all possible integer values of \( \displaystyle z \).

Find \( \displaystyle a+b+c \).

Details and Assumptions:

All the possible values of \(x,y,z\) are to be summed,not only the possible distinct values of \(x,y,z\).For example, if the solutions were \((1,2,3)\) and \((1,2,4)\),then the answer would be \(1+1+2+2+3+4=13\)

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