A8930. The Minimum Number of Variables

You've got a positive integer sequence $a_{1},a_{2},...,a_{n}$ . All numbers in the sequence are distinct. Let's fix the set of variables $b_{1},b_{2},...,b_{m}$ . Initially each variable $b_{i}$ $(1<=i<=m)$ contains the value of zero. Consider the following sequence, consisting of $n$ operations.

The first operation is assigning the value of $a_{1}$ to some variable $b_{x}$ $(1<=x<=m)$ . Each of the following $n-1$ operations is assigning to some variable $b_{y}$ the value that is equal to the sum of values that are stored in the variables $b_{i}$ and $b_{j}$ $(1<=i,j,y<=m)$ . At that, the value that is assigned on the $t$ -th operation, must equal $a_{t}$ . For each operation numbers $y,i,j$ are chosen anew.

Your task is to find the minimum number of variables $m$ , such that those variables can help you perform the described sequence of operations.

The first line contains integer $n$ $(1<=n<=23)$ . The second line contains $n$ space-separated integers $a_{1},a_{2},...,a_{n}$ $(1<=a_{k}<=10^{9})$ .

It is guaranteed that all numbers in the sequence are distinct.

In a single line print a single number — the minimum number of variables $m$ , such that those variables can help you perform the described sequence of operations.

If you cannot perform the sequence of operations at any $m$ , print -1.

In the first sample, you can use two variables $b_{1}$ and $b_{2}$ to perform the following sequence of operations.

1. $b_{1}$ := $1$ ;
2. $b_{2}$ := $b_{1}+b_{1}$ ;
3. $b_{1}$ := $b_{1}+b_{2}$ ;
4. $b_{1}$ := $b_{1}+b_{1}$ ;
5. $b_{1}$ := $b_{1}+b_{2}$ .