If the vectors $\vec{\alpha}=\hat{\mathbf{i}}+a \hat{\mathbf{j}}+\mathbf{a}^{2} \hat{\mathbf{k}}, \vec{\beta}=\hat{\mathbf{i}}+b \hat{\mathbf{j}}+\mathbf{b}^{2} \hat{\mathbf{k}}$, and $\vec{\gamma}=\hat{\mathbf{i}}+c \hat{\mathbf{j}}+c^{2} \hat{k}$ are three non-coplanar vectors and $\left|\begin{array}{lll}\mathbf{a} & \mathbf{a}^{2} & 1+\mathrm{a}^{3} \\ \mathbf{b} & \mathbf{b}^{2} & 1+\mathrm{b}^{3} \\ \mathrm{c} & \mathrm{c}^{2} & 1+\mathrm{c}^{3}\end{array}\right|=0$,
then the value of abc is