Given a set of four numbers representing a circular array we can test to see if the absolute values of the differences between each adjoining element will eventually coalesce into a single value.

For example, consider the set of four numbers: {1, 10, 8, 3}.

We can repeatedly take the absolute values of the differences of the adjoining numbers. Since this is a circular array, calculate the value of the last element in the new array as the last number minus the first number.

For example, starting with {1, 10, 8, 3}

The first iteration will produce {9, 2, 5, 2} calculated as |1 - 10|, |10 - 8|, |8 - 3|, |3 - 1| . The second iteration will produce {7, 3, 3, 7} calculated as |9 - 2|, |2 - 5|, |5 - 2|, |2 - 9|. The third iteration will produce {4, 0, 4, 0} calculated as |7 - 3|, |3 3|, |3 - 7|, |7 7|. The fourth iteration will produce {4, 4, 4, 4} calculated as |4 - 0|, |0 - 4|, |4 - 0|, |0 - 4|.

We can say the original set {1, 10, 8, 3} required 4 iterations to reach a common number (all values are 4).

In this problem you must read multiple test cases from a text file named numbers.txt.

Each line in this file will contain exactly 4 numbers separated by a single space.

Each number will be >= 0 and <= 10000.

For each set of 4 numbers, determine how many iterations are required before the sequence of the differences coalesces into a single value.

Although there are 7 lines in the test file below, do NOT assume all test files will have 7 lines of input. The files used by the judges may have more or fewer lines.

Test Data:

Input (numbers.txt)	Output
1 0 0 0	3 iterations were required for 1 0 0 0
0 1 4 11	5 iterations were required for 0 1 4 11
0 0 0 2	3 iterations were required for 0 0 0 2
34 12 19 28	4 iterations were required for 34 12 19 28
100 200 321 345	4 iterations were required for 100 200 321 345
7 2 7 2	1 iterations were required for 7 2 7 2
9934 8543 812 1001	3 iterations were required for 9934 8543 812 1001