  In the classical prophet inequality, a gambler faces a sequence of items,
whose values are drawn independently from known distributions. Upon the arrival
of each item, its value is realized and the gambler either accepts it and the
game ends, or irrevocably rejects it and continues to the next item. The goal
is to maximize the value of the selected item and compete against the expected
maximum value of all items. A tight competitive ratio of $\frac{1}{2}$ is
established in the classical setting and various relaxations have been proposed
to surpass the barrier, including the i.i.d. model, the order selection model,
and the random order model.
  In this paper, we advance the study of the order selection prophet
inequality, in which the gambler is given the extra power for selecting the
arrival order of the items. Our main result is a $0.725$-competitive algorithm,
that substantially improves the state-of-the-art $0.669$ ratio by Correa, Saona
and Ziliotto~(Math. Program. 2021), achieved in the harder random order model.
Recently, Agrawal, Sethuraman and Zhang~(EC 2021) proved that the task of
selecting the optimal order is NP-hard. Despite this fact, we introduce a novel
algorithm design framework that translates the discrete order selection problem
into a continuous arrival time design problem. From this perspective, we can
focus on the arrival time design without worrying about the threshold
optimization afterwards. As a side result, we achieve the optimal $0.745$
competitive ratio by applying our algorithm to the i.i.d. model.
