Introduction

The program below comes from a previous programming contest. The problem will give you a chance to practice manipulating arrays (both one and two-dimensional).

The Problem: Transferable Voting

The Transferable Vote system for determining the winner of an election requires that a successful candidate obtain an absolute majority of the votes cast, even when there are more than two candidates. To achieve this goal, each voter completes his or her ballot by listing all of the candidates in order of preference. Thus if there are five candidates for a single position, each voter's ballot must contain five choices, from first choice to fifth choice.

In this problem you are to determine the winner, if any, of elections using the Transferable Vote system. If there are C candidates for the single position, then each voter's ballot will include C distinct choices, corresponding to identifying the voter's first place, second place, ..., and Cth place selections. Invalid ballots will be discarded, but their presence will be noted. A ballot is invalid if a voter's choices are not distinct (choosing the same candidate as first and second choice is not permitted) or if any of the choices are not legal (that is, in the range 1 to C).

For each election candidates will be assigned sequential numbers starting with 1. The maximum number of voters in this problem will be 100, and the maximum number of candidates will be 5.

           TABLE A                  TABLE B

                                    Updated
        First  Second   Third       Choice
Voter  Choice  Choice  Choice        List
-----------------------------       ------
  1       1       2       4        
  2       1       3       2         1 3 2
  3       3       2       1         3 2 1
  4       3       2       1         3 2 1
  5       1       2       3         1 2 3
  6       2       3       1         3 1
  7       3       2       1         3 2 1
  8       3       1       1
  9       3       2       1         3 2 1
 10       1       2       3         1 2 3
 11       1       3       2         1 3 2
 12       2       3       1         3 1

Consider a small election. In Table A are the votes from 12 voters for three candidates. Voters 1 and 8 cast invalid ballots; they will not be counted. This leaves 10 valid ballots, so a winning candidate will require at least 6 votes (the least integer value greater than half the number of valid ballots) to win. Candidates 1 and 3 each have 4 first choice votes, and candidate 2 has two. There is no majority, so the candidate with the fewest first choice votes, candidate 2, is eliminated. If there had been several candidates with the fewest first choice votes, the one with the highest candidate number is eliminated. This process is repeated until a winner emerges. If all of the candidates remaining have the same number of votes, then a tie is declared.

With candidate 2 eliminated, we advance the second and third choice candidates from those votes who voted for candidate 2 as their first choice. The result of this action is shown in Table B. Now candidate 3 has picked up 2 additional votes, giving a total of 6. This is sufficient for election. Note that if voter 12 had cast the ballot "2 1 3" then there would have been a tie between candidates 1 and 3.

Input

There will be one or more elections to consider. The data for these elections should be come from a file with an appropriate name (such as election.dat). You should read, one election at a time, the data for an election from this file and print output as described below.

The data for each election must begin with a line containing the number of candidates (C), and the number of votes (N). After the last election there will be an extra data line with a 0 for the number of candidates and a 0 for the number of votes.

Following the first line for each election will be N additional lines each containing the choices from a single ballot. Each line will contain only C non-negative integers separated by whitespace.

Output

For each election print a line identifying the election number (they should be numbered sequentially starting with 1). If there were any invalid ballots, print an additional line specifying the number of invalid ballots. Finally, print a line indicating the winner of the election, if any, or an indication of a tie; be certain to identify the candidates who are tied. Separate the output for each election by a single blank line.

Sample Solution

Sample Input

3 12
1 2 4
1 3 2
3 2 1
3 2 1
1 2 3
2 3 1
3 2 1
3 1 1
3 2 1
1 2 3
1 3 2
2 3 1
3 12
1 2 4
1 3 2
3 2 1
3 2 1
1 2 3
2 3 1
3 2 1
3 1 1
3 2 1
1 2 3
1 3 2
2 1 3
4 15
4 3 1 2
4 1 2 3
3 1 4 2
1 3 2 4
4 1 2 3
3 4 2 1
2 4 3 1
3 2 1 4
3 1 4 2
1 4 2 3
3 4 1 2
3 2 1 4
4 1 3 2
3 2 1 4
4 2 1 4
0 0

Sample Output

Election #1
  2 bad ballot(s)
  Candidate 3 is elected.

Election #2
  2 bad ballot(s)
  The following candidates are tied: 1 3

Election #3
  1 bad ballot(s)
  Candidate 3 is elected.

What To Hand In

Hand in a full lab report for your program (consult the Programming Assignment Report Guidelines on what to include in this). Also include a listing of the data you used to test your program (you should discuss in your lab report what test data you used and whether this fully tests your solution). Finally you should attach output for your data showing that your program performs correctly.