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A sequence of positive rational numbers is defined as follows:
An infinite full binary tree labeled by positive rational numbers is defined by:
The top of the tree is shown in the following figure:
The 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, …
Write a program which finds the value of n for which F(n) is p/q for inputs p and q.
The first line of input contains a single integer P, (1 ≤ P ≤ 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, a single space, the numerator, p, a forward slash (/) and the denominator, q, of the desired fraction.
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 value of n for which F(n) is p/q. Inputs will be chosen so n will fit in a 32-bit integer.
4 1 1/1 2 1/3 3 5/2 4 2178309/1346269
1 1 2 4 3 11 4 1431655765