Consider the vectors \( \vec{x}=\hat{i}+2\hat{j}+3\hat{k},\quad \vec{y}=2\hat{i}+3\hat{j}+\hat{k},\; \text{and}\; \vec{z}=3\hat{i}+\hat{j}+2\hat{k}. \)
For two distinct positive real numbers \( \alpha \) and \( \beta \), define \( \vec{X}=\alpha \vec{x}+\beta \vec{y}-\vec{z},\quad \vec{Y}=\alpha \vec{y}+\beta \vec{z}-\vec{x},\; \text{and}\; \vec{Z}=\alpha \vec{z}+\beta \vec{x}-\vec{y}. \)
If the vectors \( \vec{X}, \vec{Y} \) and \( \vec{Z} \) lie in a plane, then the value of \( \alpha+\beta-3 \) is ________.
