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## D. Simple Magic

Time limit: 1.0 second
Memory limit: 64 MB
Do you think that magic is simple? That some hand-waving and muttering incomprehensible blubber is enough to conjure wonderful gardens or a fireball to burn your enemies to ashes?
The reality is a little more complicated than that. To master skills, young wizards spend years studying such subjects as magical analysis and demonology practice.
In fact, Oleg, a student of the Institute of Magic and Common Sorcery (IMCS) is preparing for an exam. And there’s no way he can calculate the Dumbledore determinant. As you might have guessed, he asked you to help him.
Let us remind you the basic definitions just in case you haven’t been visiting lectures on the theory of nonlinear spells. The Gandalf theorem states that any part of subspace can be represented as a vector of magic potentials that is an array of n positive integers. A Dumbledore determinant of this array equals the minimum number of elementary magical transformations required to turn the original array into the array where all elements are equal to one. One elementary magical transformation turns the original array of length k into a new array of length k · (k − 1) / 2. The elements of the new array are greatest common divisors of each pair of elements of the original array. For example, the elementary magical transformation of array {2, 3, 3, 6} turns it into array {gcd(2, 3), gcd(2, 3), gcd(2, 6), gcd(3, 3), gcd(3, 6), gcd(3, 6)}, that is {1, 1, 2, 3, 3, 3}.

### Input

The first line contains number n that is the length of the original array (3 ≤ n ≤ 10 000). Next n lines contain the elements of array that are positive integers not exceeding 107.

### Output

Output Dumbledore determinant for the array given in the input. If Dumbledore determinant is not defined or it exceeds 1018, output “infinity”.

### Samples

inputoutput
```3
1
2
3
```
```1
```
```4
2
2
2
2
```
```infinity
```
Problem Author: Kirill Borozdin
Problem Source: Ural Regional School Programming Contest 2013
To submit the solution for this problem go to the Problem set: 2003. Simple Magic