Let the vectors \( \vec{a}=(1+t)\hat{i}+(1-t)\hat{j}+\hat{k}, \ \vec{b}=(1-t)\hat{i}+(1+t)\hat{j}+2\hat{k} \) and \( \vec{c}=t\hat{i}-t\hat{j}+\hat{k}, \ t\in \mathbb{R} \) be such that for \( \alpha,\beta,\gamma \in \mathbb{R}, \ \alpha\vec{a}+\beta\vec{b}+\gamma\vec{c}=\vec{0} \Rightarrow \alpha=\beta=\gamma=0 \). Then, the set of all values of \( t \) is:
