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Problems

Poly-polygonal Numbers

Poly-polygonal Numbers

Time limit 1 second
Memory limit 128 MiB

A polygonal number is a number which can be represented by a regular geometrical arrangement of equally spaced points, where the arrangement forms a regular polygon. Some examples are shown in the figure below.

prb6074

The first figure shows the first 4triangular numbers 1, 3, 6, 10. The next three show the first four square, _pentagonal _and hexagonal numbers, respectively. In general, k-gonal numbers are those whose points define a regular k-gon (hence triangular numbers are 3-gonal, square numbers are 4-gonal, etc.). We will define k as an index of the polygonal number. For this problem, you are to find numbers which are k-gonal for two or more values of k. We will call these numbers poly-polygonal.

Input data

Consist of multiple problem instances. Each instance will consist of 3 lines. The first line will be a non-negative integer n50 indicating the number of types of polygonal numbers of interest in this problem. Note that this line may be longer than 80 characters. The next line will contain n integers indicating the indices of these polygonal numbers (all distinct and in increasing order). For example, if the first line contained the value 3, and the next line contained the values 3 6 10, then that problem instance would be interested in 3-gonal, 6-gonal and 10-gonal numbers. Each index k will lie in the range 3k1000. The last line of the problem instance will consist of a single positive integer s10000, which serves as a starting point for the search for poly-polygonal numbers. A value of n = 0 terminates the input.

Output data

For each problem instance, you should output the next 5 poly-polygonal numbers which are greater than or equal to s. Each number should be on a single line and conform to the following format:

num:k[1] k[2] k[3] ...

where num is the poly-polygonal number, and k[1], k[2], k[3] ... are the indices (in increasing order) of the poly-polygonal number equal to num. A single space should separate each index, and you should separate each problem instance with a single blank line. The judges input will be such that the maximum value for any poly-polygonal number will t in a long variable.

Examples

Input example #1
10
6 7 8 9 10 11 12 13 14 15
1000
5
3 4 13 36 124
1
0
Output example #1
1216:9 12
1540:6 10
1701:10 13
2300:11 14
3025:12 15

1:3 4 13 36 124
36:3 4 13 36
105:3 36
171:3 13
1225:3 4 124
Source 2000 North America, East Central Regional Contest, Problem B