Rational numbers in decimal representation can have an infinite periodic part. One common way to write this down is to repeat the periodic digits and then add three dots. Numbers without those three dots are taken to be non-periodic (or equivalently, have a periodic `0` after the given digits). Your task is to decode this representation into a fully cancelled fraction.
In particular, the numbers to decode are given as follows:
* There is an optional sign (+ or -). If omitted, + is assumed.
* There is an integral part. If empty, it is taken
to be 0.
* There is a decimal point, which is optional if the number
is an integer and the integral part was not omitted.
* There is an fractional part following the integer.
* If the fractional part has at least one digit, it may be
followed by three dots.
The program shall take a string as input, and give a fully cancelled fraction (as pair numerator/denominator) as result. It may assume that the given string conforms to the number format.
The string has to be interpreted as follows:
* If the string does not end in three dots, it is interpreted
as exact number.
* If the string does end in three dots, find the longest
repeating digit sequence in the fractional part preceding
the three dots. If no such repeating sequence is found,
the period consists of just the last digit. Otherwise
it consists of that longest repeating digit sequence.
* Output the fraction that corresponds to the determined
periodic digit sequence. The denominator shall be
positive. The fraction shall be completely cancelled.
* Your code must handle at least up to 6 digits before and
up to 6 digits after the decimal point.
Test cases:
```
"0" -> 0/1
"-0" -> 0/1
"+0.0" -> 0/1
"42" -> 42/1
"+2" -> 2/1
"-6" -> -6/1
"-2.0" -> -2/1
"0815" -> 815/1
"." -> 0/1
"+." -> 0/1
"-." -> 0/1
".0" -> 0/1
"+00.0" -> 0/1
"-.2" -> -1/5
"3.14" -> 157/50
".3..." -> 1/3
"+.11..." -> 1/9
"1.0..." -> 1/1
"2.9..." -> 3/1
"0.121..." -> 109/900
"0.1212..." -> 4/33
"0.12121..." -> 4/33
"0.12122..." -> 1091/9000
".122122..." -> 122/999
".022122..." -> 1991/90000
".0221221..." -> 221/9990
"-123456.123434..." -> -611107811/4950
```
