Biology

Vera has $A \times B$ cards. Each card has a rank, an integer between $0$ and $A - 1$, and a suit, an integer between $0$ and $B - 1$. All cards are distinct. A set of five different cards is a hand.

Every hand falls into exactly one of nine categories numbered $1$ through $9$. If a hand satisfies the conditions of more than one category, it belongs to the lowest numbered category among them. The rules are:

1. Straight flush: the hand is a straight and a flush.
2. Four of a kind: four of the cards have the same rank.
3. Full house: three of the cards have the same rank and the remaining two have the same rank.
4. Flush: all five cards have the same suit.
5. Straight: the ranks of the cards in increasing order are $x, x + 1, x + 2, x + 3, x + 4$ for some integer $x$.
6. Three of a kind: three of the cards have the same rank.
7. Two pair: two cards have the same rank and two other cards have the same rank.
8. One pair: two cards have the same rank.
9. High card: the hand satisfies no other category.

Vera currently holds two cards with ranks $a_1$, $a_2$ and suits $b_1$, $b_2$. She picks three more cards from the remaining ones and forms a hand with the two she holds. Compute the number of different hands formed this way that belong to each category.

The first line contains the integers $A$ and $B$ ($5 \le A \le 25$, $1 \le B \le 4$).

The second line contains the integers $a_1$, $b_1$, $a_2$, $b_2$ ($0 \le a_1, a_2 \le A - 1$, $0 \le b_1, b_2 \le B - 1$, $(a_1, b_1) \neq (a_2, b_2)$).

Print one line with nine integers separated by single spaces. The $i$-th integer is the number of different hands that belong to category $i$, so the counts run from straight flush to high card in increasing order of category number.

Let $(a, b)$ denote the card with rank $a$ and suit $b$.

In the first example Vera holds $(1, 0)$ and $(3, 1)$. If she picks $(3, 0)$, $(4, 0)$, $(4, 1)$, her hand has two cards of rank $3$ and two other cards of rank $4$, so it is a two pair. The same hand also satisfies one pair, but two pair has the lower category number, so it counts as a two pair.