Juggle with Criteria

문제

A permutation p of length n is an array {p₁, p₂, . . . , p_n} that contains each integer from 1 to n once. Jack devised five criteria that are supposed to show how close a permutation p is to the identity permutation {1, 2, . . . , n}:

• a(p) = the number of inversions in p (pairs of indices (i, j) such that i < j and p_i > p_j );
• b(p) = the number of local inversions in p (indices i such that p_i > p_i+1);
• c(p) = the length of longest increasing subsequence in p (p_i1 < p_i2 < . . . < p_ik with increasing indices i₁ < i2 < . . . < ik);
• d(p) = the length of longest increasing substring in p (increasing consecutive subsequence p_i < p_i+1 < . . . < p_j−1 < p_j );
• e(p) = the number of fixed points (indices i such that p_i = i).

Jill wants to prove him that these criteria can vary independently (in some sense). She need to show that two permutations p and q of equal length can give any combination of:

• whether a(p) is less, equal, or greater than a(q);
• whether b(p) is less, equal, or greater than b(q);
• whether c(p) is less, equal, or greater than c(q);
• whether d(p) is less, equal, or greater than d(q); and
• whether e(p) is less, equal, or greater than e(q).

Help Jill prove her thesis in a constructive way. For a given set of relations between these criteria, find two permutations p and q of the same length that satisfy these relations.

입력

The first line of the input file contains two integer numbers n and l — the number of relations and permutations length (1 ≤ n ≤ 243; 1 ≤ l ≤ 1000).

Each of the following n lines contains one set criteria that is defined by the five characters. Each character is either ‘<’, ‘=’, or ‘>’. These characters denote desired relations between a(p) and a(q), . . ., e(p) and e(q) (in this order)

출력

For each set of criteria given in the input file output either “Exists” if there exists a pair of permutations p and q of length l satisfying this set of criteria, and “Not exists” otherwise.

In the former case following two lines must contain permutation p and q (in this order).

힌트

In the first pair of permutations in sample output:

a(p)= 2 < 3=a(q); b(p)= 1 = 1=b(q); c(p)= 3 = 3=c(q); d(p)= 2 < 3=d(q); e(p)= 1 > 0=e(q).

3 4
<==<>
<<<<<
=====

Exists
1 4 2 3
2 3 4 1
Not exists
Exists
1 2 3 4
1 2 3 4