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2017-04-15 19:00 CEST

NAIPC 2017

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2017-04-16 00:00 CEST
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Problem C
Stretching Streamers

Ms. Hall wants to teach her class about common factors. She arranges her students in a circle and assigns each student an integer in the range $[2,10^9]$. She also provides the students with crepe paper streamers. The students are to stretch these streamers between pairs of students, and pull them tight. But, there are some rules.

  • Two students can stretch a streamer between them if and only if their assigned integers share a factor other than $1$.

  • There is exactly one path, going from streamer to streamer, between any two students in the circle.

  • No streamers may cross.

  • Any given student may hold an end of arbitrarily many streamers.

Suppose Ms. Hall has four students, and she gives them the numbers $2$, $3$, $30$ and $45$. In this arrangement, there is one way to stretch the streamers:

\includegraphics[width=.2\textwidth ]{Slide1.JPG}

In this arrangement, there are three ways to stretch the streamers:

\includegraphics[width=.2\textwidth ]{Slide2.JPG} \includegraphics[width=.2\textwidth ]{Slide3.JPG} \includegraphics[width=.2\textwidth ]{Slide4.JPG}

In how many ways can the students hold the streamers subject to Ms. Hall’s rules? Two ways are different if and only if there is a streamer between two given students one way, but none between those two students the other way.

Input

Each input will consist of a single test case. Note that your program may be run multiple times on different inputs. The first line of input will contain a single integer $n$ ($2 \le n \le 300$), which is the number of Ms. Hall’s students.

Each of the next $n$ lines will hold an integer $x$ ($2 \le x \le 10^9$). These are the numbers held by the students, in order. Remember, the students are standing in a circle, so the last student is adjacent to the first student.

Output

Output a single integer, which is the number of ways Ms. Hall’s students can satisfy her rules. Since this number may be very large, output it modulo $10^9+7$.

Sample Input 1 Sample Output 1
4
30
3
2
45
1
Sample Input 2 Sample Output 2
4
3
30
2
45
3