CF1295F Good Contest

An online contest will soon be held on ForceCoders, a large competitive programming platform. The authors have prepared $ n $ problems; and since the platform is very popular, $ 998244351 $ coder from all over the world is going to solve them. For each problem, the authors estimated the number of people who would solve it: for the $ i $ -th problem, the number of accepted solutions will be between $ l_i $ and $ r_i $ , inclusive. The creator of ForceCoders uses different criteria to determine if the contest is good or bad. One of these criteria is the number of inversions in the problem order. An inversion is a pair of problems $ (x, y) $ such that $ x $ is located earlier in the contest ( $ x < y $ ), but the number of accepted solutions for $ y $ is strictly greater. Obviously, both the creator of ForceCoders and the authors of the contest want the contest to be good. Now they want to calculate the probability that there will be no inversions in the problem order, assuming that for each problem $ i $ , any integral number of accepted solutions for it (between $ l_i $ and $ r_i $ ) is equally probable, and all these numbers are independent.

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The real answer in the first test is $ \frac{1}{2} $ .