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Let \( \omega \) be a complex number such that \( 2 \omega+1=z \)
\( \mathrm{P} \) where \( z=\sqrt{-3} \). If \( \left|\begin{array}{ccc}1 & 1 & 1 \\ 1 & -\omega^{2}-1 & \omega^{2} \\ 1 & \omega^{2} & \omega^{7}\end{array}\right|=3 k \) then \( k \) is equal to:
(1) \( z \)
(2) -1
(3) 1
(4) \( -z \)
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