ICPC East Central NA Regional Contest 2000, Problem B

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.

The first figure shows the first 4 triangular 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 Specification

Input will consist of multiple problem instances. Each instance will consist of 3 lines. The first line will be a non-negative integer $n\le 50$ 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 $3610$, then that problem instance would be interested in 3-gonal, 6-gonal and 10-gonal numbers. Each index $k$ will lie in the range $3\le k\le 1000$. The last line of the problem instance will consist of a single positive integer $s\le 10000$, which serves as a starting point for the search for poly-polygonal numbers. A value of $n=0$ terminates the input.

Output Specification

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:k1 k2 k3 ...

where num is the poly-polygonal number, and k1, k2, k3 … 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 judge's input will be such that the maximum value for any poly-polygonal number will fit in a long variable.

Sample Input

10
6 7 8 9 10 11 12 13 14 15
1000
5
3 4 13 36 124
1
0

Sample Output

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