On a question of Bellman

From a random point $O$ in an infinite strip of width 1 we move in a randomly chosen direction along a curve $\Gamma$. What shape of $\Gamma$ gives the minimum value to the expectation of the length of the path that reaches the boundary of the strip? After certain arguments in support that the desired curve belongs to one of four classes, it is proved that the best curve in those classes consists of four parts: an interval $OA$ of length $a$, its smooth continuation, an arc $AB$ of radius 1 and small length $\varphi$, an interval $BD$ that is smooth continuation of the arc, and an interval $DF$ (with a corner at the point $D$). If we take $O$ as the origin and $OA$ as the $x$ axis of a system of coordinates, then the coordinates of the above-mentioned points will be: $A(a,0)$, $B(a+\sin\varphi,1-\cos\varphi)$, $F(a,1)$,
$$ D\left(a+\frac{\cos\varphi\sqrt{1+a^2}-a}{\cos\varphi-a \sin\varphi}, 1-\frac{\sin\varphi\sqrt{1+a^2}}{\cos\varphi-a\sin\varphi}\right). $$
For the best curve we have: $a\approx0.814$, $\varphi\approx0.032$. Related questions are discussed.