CF327E Axis Walking

Iahub wants to meet his girlfriend Iahubina. They both live in $ Ox $ axis (the horizontal axis). Iahub lives at point 0 and Iahubina at point $ d $ . Iahub has $ n $ positive integers $ a_{1} $ , $ a_{2} $ , ..., $ a_{n} $ . The sum of those numbers is $ d $ . Suppose $ p_{1} $ , $ p_{2} $ , ..., $ p_{n} $ is a permutation of $ {1,2,...,n} $ . Then, let $ b_{1}=a_{p1} $ , $ b_{2}=a_{p2} $ and so on. The array b is called a "route". There are $ n! $ different routes, one for each permutation $ p $ . Iahub's travel schedule is: he walks $ b_{1} $ steps on $ Ox $ axis, then he makes a break in point $ b_{1} $ . Then, he walks $ b_{2} $ more steps on $ Ox $ axis and makes a break in point $ b_{1}+b_{2} $ . Similarly, at $ j $ -th $ (1

The first line contains an integer $ n $ ( $ 1

Output a single integer — the answer of Iahub's dilemma modulo $ 1000000007 $ ( $ 10^{9}+7 $ ).

In the first case consider six possible orderings: - \[2, 3, 5\]. Iahub will stop at position 2, 5 and 10. Among them, 5 is bad luck for him. - \[2, 5, 3\]. Iahub will stop at position 2, 7 and 10. Among them, 7 is bad luck for him. - \[3, 2, 5\]. He will stop at the unlucky 5. - \[3, 5, 2\]. This is a valid ordering. - \[5, 2, 3\]. He got unlucky twice (5 and 7). - \[5, 3, 2\]. Iahub would reject, as it sends him to position 5. In the second case, note that it is possible that two different ways have the identical set of stopping. In fact, all six possible ways have the same stops: \[2, 4, 6\], so there's no bad luck for Iahub.