Problem Description
Consider an infinite complete binary tree where the root node is 1/1 and left and right childs of node p/q are p/(p+q) and (p+q)/q, respectively. This tree looks like:
1/1 ______|______ | | 1/2 2/1 ___|___ ___|___ | | | | 1/3 3/2 2/3 3/1 ...
It is known that every positive rational number appears exactly once in this tree. A level-order traversal of the tree results in the following array:
1/1, 1/2, 2/1, 1/3, 3/2, 2/3, 3/1, ...
Please solve the following two questions:
- Find the n-th element of the array, where n starts from 1. For example, for the input 2, the correct output is 1/2.
- Given p/q, find its position in the array. As an example, the input 1/2 results in the output 2.
Input
The first line of the input gives the number of test cases, T(1 ≤ T ≤ 100).
T test cases follow. Each test case consists of one line.
The line contains a problem id (1 or 2) and one or two additional integers:
- If the problem id is 1, then only one integer n is given, and you are expected to find the n-th element of the array.
- If the problem id is 2, then two integers p and q are given, and you are expected to find the position of p/q in the array.
p and q are relatively prime.
1 ≤ n, p, q ≤ 264-1
p/q is an element in a tree with level number ≤ 64.
Output
For each test case:
- If the problem id is 1, then output one line containing "Case #x: p q", where x is the case number (starting from 1), and p, q are numerator and denominator of the asked array element, respectively.
- If the problem id is 2, then output one line containing "Case #x: n", where x is the case number (starting from 1), and n is the position of the given number.
Sample Input
4 1 2 2 1 2 1 5 2 3 2
Sample Output
Case #1: 1 2 Case #2: 2 Case #3: 3 2 Case #4: 5
