E. Ehab and the Expected GCD Problemtime limit per test2 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputLet's define a function f(p) on a permutation p as follows. Let g_i be the greatest common divisor (GCD) of elements p_1, p_2, ..., p_i (in other words, it is the GCD of the prefix of length i). Then f(p) is the number of distinct elements among g_1, g_2, ..., g_n.Let f_{max}(n) be the maximum value of f(p) among all permutations p of integers 1, 2, ..., n.Given an integers n, count the number of permutations p of integers 1, 2, ..., n, such that f(p) is equal to f_{max}(n). Since the answer may be large, print the remainder of its division by 1000\,000\,007 = 10^9 + 7.InputThe only line contains the integer n (2 \le n \le 10^6) — the length of the permutations.OutputThe only line should contain your answer modulo 10^9+7.ExamplesInput
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120NoteConsider the second example: these are the permutations of length 3:  [1,2,3], f(p)=1.  [1,3,2], f(p)=1.  [2,1,3], f(p)=2.  [2,3,1], f(p)=2.  [3,1,2], f(p)=2.  [3,2,1], f(p)=2. The maximum value f_{max}(3) = 2, and there are 4 permutations p such that f(p)=2.
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['combinatorics', 'dp', 'math', 'number theory', '*2500']