Problem B. Rational Number Tree

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Small input
9 points

The contest is finished.

Large input
12 points

The contest is finished.

Problem

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:

1. 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.
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. 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:

1. 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.
2. 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.

Output

For each test case:

1. 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.
2. 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.

Limits

1 ≤ T ≤ 100; p and q are relatively prime.

Small dataset

1 ≤ n, p, q ≤ 2¹⁶-1; p/q is an element in a tree with level number ≤ 16.

Large dataset

1 ≤ n, p, q ≤ 2⁶⁴-1; p/q is an element in a tree with level number ≤ 64.

Sample

Input 
 
Output 
 

41 22 1 21 52 3 2

Case #1: 1 2Case #2: 2Case #3: 3 2Case #4: 5