Coordinate Geometry: JEE Main 2025 Mathematics question

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 value(s) of $\lambda$ is: