The mass and stiffness matrices and the mode shapes of a two-degree-of-freedom system are given by

\[[m]=\left[\begin{array}{ll}1 & 0 \\0 & 4\end{array}\right], \quad[k]=\left[\begin{array}{cc}12 & -k_{12} \\-k_{12} & k_{22}\end{array}\right], \quad \vec{X}^{(1)}=\left\{\begin{array}{c}1 \\9.1109\end{array}\right\}, \quad \vec{X}^{(2)}=\left\{\begin{array}{c}-9.1109 \\1\end{array}\right\}\]

If the first natural frequency is given by \(\omega_{1}=1.7000\), determine the stiffness coefficients \(k_{12}\) and \(k_{22}\) and the second natural frequency of vibration, \(\omega_{2}\).