Ehab and the Expected GCD Problem

$p$ 是一个排列，定义 $f(p)$ ： 设 $g_i$ 为 $p_1, p_2, \cdots, p_i$ 的最大公因数（即前缀最大公因数），则 $f(p)$ 为 $g_1,g_2,\cdots,g_n$ 中不同的数的个数。 设 $f_{max}(n)$ 为 $1,2,\cdots,n$ 的所有排列 $p$ 中 $f(p)$ 的最大值，你需要求有多少 $1,2,\cdots,n$ 的排列 $p$ 满足 $f(p)=f_{max}(n)$，答案对 $10^9+7$ 取模。

Let's define a function $ f(p) $ on a permutation $ p $ as follows. Let $ g_i $ be the [greatest common divisor (GCD)](https://en.wikipedia.org/wiki/Greatest_common_divisor) 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 $ .

The only line contains the integer $ n $ ( $ 2 \le n \le 10^6 $ ) — the length of the permutations.

The only line should contain your answer modulo $ 10^9+7 $ .

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120

Consider 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 $ .