Veshy needs help in math class. He has sequences of 10 spaced terms in order of . For each sequence he wants to know if it is arithmetic, geometric, or neither. Output the answer to the sequence on the line. Terms are guaranteed to be integers.

Note:

A arithmetic sequence is a sequence such that it can be written in the form: where and are constants.

A geometric sequence is a sequence such that it can be written in the form: where and are constants.

**It may be helpful to know that in an arithmetic sequence, and in a geometric sequence, **

#### Constraints

In all tests,

#### Input Specification

The first line of input is .

Each the following lines contains 10 integers, , a sequence of numbers.

#### Output Specification

Your output must have lines such that the answer to the sequence is on the line.

If the sequence is arithmetic, output `arithmetic`

.

If the sequence is geometric, output `geometric`

.

If the sequence is neither arithmetic nor geometric, output `neither`

.

If the sequence is both arithmetic and geometric, output `both`

.

#### Sample Input

```
4
1 2 3 4 5 6 7 8 9 10
2 4 8 16 32 64 128 256 512 1024
1 1 0 0 1 1 0 0 1 1
1 1 1 1 1 1 1 1 1 1
```

#### Sample Output

```
arithmetic
geometric
neither
both
```

## Comments