A Rational Sequence
An infinite full binary tree labeled by positive rational numbers is defined by:
The label of the root is $1/1$.
The left child of label $p/q$ is $p/(p+q)$.
The right child of label $p/q$ is $(p+q)/q$.
The top of the tree is shown in the following figure:
A rational sequence is defined by doing a level order (breadth first) traversal of the tree (indicated by the light dashed line). So that:\[ F(1) = 1/1, F(2) = 1/2, F(3) = 2/1, F(4) = 1/3, F(5) = 3/2, F(6) = 2/3, \ldots \]
Write a program which takes as input a rational number, $p/q$, in lowest terms and finds the next rational number in the sequence. That is, if $F(n) = p/q$, then the result is $F(n+1)$.
The first line of input contains a single integer $P$, ($1 \le P \le 1000$), which is the number of data sets that follow. Each data set should be processed identically and independently.
Each data set consists of a single line of input. It contains the data set number, $K$, which is then followed by a space, then the numerator of the fraction, $p$, followed immediately by a forward slash (/), followed immediately by the denominator of the fraction, $q$. Both $p$ and $q$ will be relatively prime and $0 \le p, q \le 2\, 147\, 483\, 647$.
For each data set there is a single line of output. It contains the data set number, $K$, followed by a single space which is then followed by the numerator of the fraction, followed immediately by a forward slash (/) followed immediately by the denominator of the fraction. Inputs will be chosen such that neither the numerator nor the denominator will overflow a 32-bit integer.
|Sample Input 1||Sample Output 1|
5 1 1/1 2 1/3 3 5/2 4 2178309/1346269 5 1/10000000
1 1/2 2 3/2 3 2/5 4 1346269/1860498 5 10000000/9999999