### Problem Statement

A power series takes the form:
```  f(x) =  a0 + a1x + a2x2 + a3x3 + ...
```
If we had another power series
```  g(x) =  b0 + b1x + b2x2 + b3x3 + ...
```
then we compute the product as follows:
`  f(x)g(x) = a0b0 + (a0b1 + a1b0)x + (a0b2 + a1b1 + a2b0)x2 + ...`
More formally, we have
```   ck = sum (i = 0 to k) aibk-i
```
where ck is a coefficient of the power series for the product.

Given the series f(x) you must determine g(x) such that f(x)g(x) = 1 + 0x + 0x2 + ... The first few a-values will be given in a int[] start. The remaining a-values will be an infinitely repeating sequence of the terms in repeat. For example, if start = { 1, 2 } and repeat = { 3, 4, 5 } then the resulting power series is
```    1 + 2x + 3x2 + 4x3 + 5x4 + 3x5 + 4x6 + 5x7 + ...
```
You will return the first n terms of g(x) as a String[] where the first element is b0, the second is b1 and so forth. Each bi should be given in the form "p/q" where p and q are integers with no common factors (other than 1), and q is positive. If bi equals 0, then use the string "0/1". Neither p nor q should have extra leading zeros, and if p is negative, it should have a single leading '-'.

### Definition

 Class: CauchyProduct Method: findInverse Parameters: int[], int[], int Returns: String[] Method signature: String[] findInverse(int[] start, int[] repeat, int n) (be sure your method is public)

### Constraints

-start will contain between 1 and 50 elements, inclusive.
-repeat will contain between 1 and 50 elements, inclusive.
-Each element of start will be between -20 and 20, inclusive.
-Each element of repeat will be between -20 and 20, inclusive.
-Element 0 of start will not be 0.
-n will be between 1 and 1000, inclusive.
-Each integer in each element of the return value will be between -100000 and 100000, inclusive.

### Examples

0)

 `{1} ` `{1}` `5`
`Returns: {"1/1", "-1/1", "0/1", "0/1", "0/1" }`
1)

 `{1}` `{2}` `5`
`Returns: {"1/1", "-2/1", "2/1", "-2/1", "2/1" }`
2)

 `{1,2}` `{3,4,5}` `20`
```Returns:
{"1/1",
"-2/1",
"1/1",
"0/1",
"0/1",
"3/1",
"-9/1",
"9/1",
"0/1",
"-9/1",
"18/1",
"-36/1",
"45/1",
"-9/1",
"-63/1",
"126/1",
"-171/1",
"180/1",
"-36/1",
"-333/1" }```
3)

 `{1,2} ` `{3,4,5}` `35`
```Returns:
{"1/1",
"-2/1",
"1/1",
"0/1",
"0/1",
"3/1",
"-9/1",
"9/1",
"0/1",
"-9/1",
"18/1",
"-36/1",
"45/1",
"-9/1",
"-63/1",
"126/1",
"-171/1",
"180/1",
"-36/1",
"-333/1",
"747/1",
"-927/1",
"720/1",
"99/1",
"-1818/1",
"3960/1",
"-4923/1",
"3123/1",
"2097/1",
"-10674/1",
"20457/1",
"-24552/1",
"13464/1",
"17379/1",
"-62865/1" }```
4)

 `{2,3,4}` `{5,6,7}` `5`
`Returns: {"1/2", "-3/4", "1/8", "1/16", "1/32" }`
5)

 `{-19,14,-8,-2,-19,12,-4}` ```{-6,-6,2,14,-5,-18,17,-8,8, -16,-2,-12,11,16,-12,-17,5, 1,11,11,0,10,-14,8,1,4, -8,1,-16,16,19,-4,18,4, -11,-15,9,4,-8,10,-5,-9,9,-9,16,10}``` `3`
`Returns: {"-1/19", "-14/361", "-44/6859" }`

#### Problem url:

http://www.topcoder.com/stat?c=problem_statement&pm=6778

#### Problem stats url:

http://www.topcoder.com/tc?module=ProblemDetail&rd=10842&pm=6778