The problem is a improved version of this problem, A random walk with uniformly distributed steps

Let us imagine a point on the real axis. At the beginning, it is located at point $O$. Then it will "walk" on the real axis randomly. For every step of the "walk", it will choose a real number $\Delta x$ in interval $[l,r]$ equiprobably, and turn right and move $\Delta x$ unit. Once it move to the left side of the point $O$, it will "die" immediately.

Our task is find out the probability of the point "live" after n steps of "walk" $P_n$. I have tried to solve it and found out a method to count $P_n$ with $\Theta (n^4)$ of time complexity, using fourier transform and something in complex analysis. But is there a more simple method? Or is there one which needs lower time complexity?