Mathematics Question from JEE Main 2025 · 29 January, Shift 2

Let the line $x + y = 1$ meet the axes of $x$ and $y$ at A and B, respectively. A right-angled triangle AMN is inscribed in the triangle OAB, where O is the origin and the points M and N lie on the lines OB and AB, respectively. If the area of the triangle AMN is $\frac{4}{9}$ of the area of the triangle OAB and $AN : NB = \lambda : 1$, then the sum of all possible values of $\lambda$ is: