If the vectors, $\vec{p}=(a+1) \hat{i}+a \hat{j}+a \hat{k}$, $\overrightarrow{\mathrm{q}}=\mathrm{a} \hat{\mathrm{i}}+(\mathrm{a}+1) \hat{\mathrm{j}}+\mathrm{a} \hat{\mathrm{k}}$, and $\vec{r}=a \hat{i}+a \hat{j}+(a+1) \hat{k}(a \in R)$ are coplanar and $3(\vec{p} \cdot \vec{q})^2-\lambda|\vec{r} \times \vec{q}|^2=0$, then the value of $\lambda$ is $\_\_\_\_$ .