P6274 [eJOI 2017] Six

Elly is studying some properties of a positive integer $N$, and she finds that $N$ has no more than $6$ distinct prime factors. Next, she generates a list in a special way. The list is empty at the beginning. Each time, she writes down a divisor $x$ of $N$ that is greater than $1$ and adds it to the list. Before adding $x$, she must ensure that among all numbers already in the list, there are **no more than** $1$ number that is **not coprime** with $x$. ------------------------------ For example, when $N=12156144$: Valid lists include: $ (42), (616, 6, 91, 23),(91, 616, 6, 23), (66, 7), (66, 7, 7, 23, 299, 66), \\(143, 13, 66),(42,12156144),\text{etc.} $ Invalid lists include: $(5,11)$, because $5$ is not a divisor of $N$; and $ (66, 13, 143)$, because $143$ is not coprime with both of the other numbers. ----------------------- Now Elly wants you to compute, for a given $N$, how many different valid lists can be generated. **We say two lists are different if their lengths are different, or if there exists a position where the values at that position are different.**

One integer: $N$.

One integer, the number of lists modulo $(10^9+7)$.

#### 【Explanation of Samples】 **Sample 1 Explanation** The lists that satisfy the condition are: $(2), (2, 2), (2, 2, 3), (2, 2, 3, 3), (2, 3), (2, 3, 2), (2, 3, 2, 3), (2, 3, 3), (2, 3, 3, 2), \\ (2, 6), (2, 6,3), (3), (3, 2), (3, 2, 2), (3, 2, 2, 3), (3, 2, 3), (3, 2, 3, 2), (3, 3),\\ (3, 3, 2), (3, 3, 2,2), (3, 6), (3, 6, 2), (6), (6, 2), (6, 2, 3), (6, 3), (6, 3, 2), (6, 6)$. There are $28$ lists in total. **Sample 4 Explanation** The real answer is $14104757650$. Output $14104757650 \bmod (10^9+7)=104757552$. #### 【Constraints and Notes】 - For all testdata, it is guaranteed that $1\le N\le 10^{15}$. - For about $30\%$ of the testdata, $N$ has at most $2$ prime factors. - For about $60\%$ of the testdata, $N$ has at most $4$ prime factors. - For $100\%$ of the testdata, $N$ has at most $6$ prime factors. #### 【Notes】 The original problem is from: [eJOI 2017](www.ejoi.org) Problem B [Six](http://ejoi.org/wp-content/themes/ejoi/assets/pdfs/tasks_day_1/EN/six_statement-en.pdf). Translation provided by: @[```_Wallace_```](https://www.luogu.com.cn/user/61430). Translated by ChatGPT 5