Ellipse

Let the line \(2 x+3 y-\mathrm{k}=0, \mathrm{k}>0\), intersect the \(x\)-axis and \(y\)-axis at the points \(\mathrm{A}\) and \(\mathrm{B}\), respectively. If the equation of the circle having the line segment \(A B\) as a diameter is \(x^2+y^2-3 x-2 y=0\) and the length of the latus rectum of the ellipse \(x^2+9 y^2=k^2\) is \(\frac{m}{n}\), where \(m\) and \(n\) are coprime, then \(2 \mathrm{~m}+\mathrm{n}\) is equal to