\[ \begin{cases} x_1 +x_3=1 \\ -x_2+2x_3=\omega \\ x_1 - x_2 + 3x_3=2 \\ x_1+x_2-x_3 =1-\omega \end{cases} \]

Consider the linear systems above, where \(\omega\) is an arbitrary real constant.

Determine the value(s) of \(\omega\) that results in a consistent system.

**Note**: If there's more than one value of \(\omega \), then submit their sum as your answer.

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