Parabola

Let the tangent to the curve \(x^{2}+2 x-4 y+9=0\) at the point \(\mathrm{P}(1,3)\) on it meet the \(y\)-axis at \(\mathrm{A}\). Let the line passing through \(\mathrm{P}\) and parallel to the line \(x-3 y=6\) meet the parabola \(y^{2}=4 x\) at \(\mathrm{B}\). If \(\mathrm{B}\) lies on the line \(2 x-3 y=8\), then \((\mathrm{AB})^{2}\) is equal to ___________.