Coordinate Geometry: JEE Main 2024 Mathematics question

Let $L_1, L_2$ be the lines passing through the point $P(0,1)$ and touching the parabola $9 x^2+12 x+18 y-14=0$. Let $Q$ and $R$ be the points on the lines $L_1$ and $L_2$ such that the $\triangle P Q R$ is an isosceles triangle with base $Q R$. If the slopes of the lines $Q R$ are $m_1$ and $m_2$, then $16\left(m_1^2+m_2^2\right)$ is equal to __________.