Ant Amelia starts on the number line at $0$ and crawls in the following manner. For $n=1,2,3$, Amelia chooses a time duration $t_n$ and an increment $x_n$ independently and uniformly at random from the interval $(0,1)$. During the $n$th step of the process, Amelia moves $x_n$ units in the positive direction, using up $t_n$ minutes. If the total elapsed time has exceeded $0.5$ minutes during the $n$th step, she stops at the end of that step. Otherwise, she continues with the next step, taking at most $3$ steps in all. What is the probability that Amelia’s position when she stops will be greater than $0.5$?