_To be frank with you, this problem is an extension of a classic problem, of which the colossal data range might increase its difficulty._
Let's define a type of operation with respect to a positive integer $n$ as replacing it with one of its positive factor $d$.
Given the integer $n$ ($n > 1$), you are asked to determine the expected times of doing this operation repeatedly until $n$ is changed into $1$ for the first time, assuming that $d$ is selected with equal probability among all the possible factors at any time.
For ease of calculation, $n$ and all its distinct prime factors $p_1$, $p_2$, $\\dots$, $p_m$ will be given.
To avoid the precision issue, let's express the expected value as a reduced fraction $frac(a, b)$, and you should output the minimum non-negative integer $c$ such that $b c equiv a pmod 10^9 + 7$.
The input contains multiple (about $2 times 10^5$) test cases.
For each test case, the first line contains two integers $n$, $m$ ($2 <= n <= 10^(24)$, $m >= 1$), where $m$ indicates the number of distinct prime factors of $n$.
The second lines contains $m$ distinct prime numbers $p_1, p_2, \\dots, p_m$ ($2 <= p_1, p_2, \\dots, p_m <= 10^6$).
For each test case, output "_Case #x: y_" in one line (without quotes), where $x$ indicates the case number starting from $1$, and $y$ denotes the answer to the corresponding case.
It is guaranteed that the answer exists for each test case.
## Input
The input contains multiple (about $2 times 10^5$) test cases.For each test case, the first line contains two integers $n$, $m$ ($2 <= n <= 10^(24)$, $m >= 1$), where $m$ indicates the number of distinct prime factors of $n$.The second lines contains $m$ distinct prime numbers $p_1, p_2, \\dots, p_m$ ($2 <= p_1, p_2, \\dots, p_m <= 10^6$).
## Output
For each test case, output "_Case #x: y_" in one line (without quotes), where $x$ indicates the case number starting from $1$, and $y$ denotes the answer to the corresponding case.It is guaranteed that the answer exists for each test case.
[samples]
**Definitions**
Let $ n \in \mathbb{Z} $ be the number of words.
Let $ W = (w_1, w_2, \dots, w_n) $ be the sequence of words, where each $ w_i $ is a string over lowercase English letters.
**Constraints**
1. $ 1 \leq n \leq 8 \times 10^6 $
2. $ \sum_{i=1}^{n} |w_i| \leq 8 \times 10^6 $
3. Each $ w_i $ contains only lowercase English letters.
**Objective**
Identify the set of repeated words as follows:
A word $ w_i $ is *repeated* if it has appeared earlier in the sequence (i.e., $ \exists j < i $ such that $ w_j = w_i $) **and** $ |w_i| \geq 4 $.
Let $ R $ be the list of such repeated words, in the order of their *second* (and subsequent) occurrence.
Output:
- If $ R $ is empty: print `"SAFO"`
- Otherwise:
1. Print $ |R| $
2. Print all words in $ R $, separated by spaces