Let $A=\left[\begin{array}{l}a_1 \\ a_2\end{array}\right]$ and $B=\left[\begin{array}{l}b_1 \\ b_2\end{array}\right]$ be two $2 \times 1$ matrices with real entries such that $\mathrm{A}=\mathrm{XB}$, where $X=\frac{1}{\sqrt{3}}\left[\begin{array}{cc}1 & -1 \\ 1 & k\end{array}\right]$, and $k \in R$. If $$ \mathrm{a}_1^2+\mathrm{a}_2^2=\frac{2}{3}\left(\mathrm{~b}_1^2+\mathrm{b}_2^2\right) \text { and }\left(\mathrm{k}^2+1\right) \mathrm{b}_2^2 \neq-2 \mathrm{~b}_1 \mathrm{~b}_2 $$ Then the value of k is $\_\_\_\_$ .