Believer in I 2

Hyeonjong joined the Order of I, a group that treats the number $I$ as sacred. The order calls $I$ a good number, and it calls every other number you can build from copies of $I$ with arithmetic a good number too. To make many good numbers, Hyeonjong plays the following game.

He prepares two things.

• $A$ cards with $I$ drawn on them, $B$ cards with $+$, and $C$ cards with $\times$. Cards that carry the same symbol look alike, so he cannot tell them apart.
• A stack that already holds infinitely many copies of $I$.

Hyeonjong lays the $A+B+C$ cards in a row and draws them from left to right. Each drawn card triggers one action.

• $I$ card: push $I$ onto the stack.
• $+$ card: pop the top two numbers of the stack, then push their sum.
• $\times$ card: pop the top two numbers of the stack, then push their product.

The stack holds infinitely many copies of $I$, so a pop never runs out of numbers.

Two rows of cards differ only when the sequence of symbols differs, so there are $\frac{(A+B+C)!}{A!B!C!}$ rows in total. Hyeonjong runs the whole procedure for every possible row, and for each $i$ from 1 to $K$ he wants the sum of the numbers that sit $i$-th from the top of the final stack. Help him compute these $K$ sums.

The first line contains five integers $I$, $A$, $B$, $C$, $K$ separated by spaces. ($1 \le I \le 10^9$, $0 \le A, B, C \le 40$, $1 \le K \le 40$)

$A$ is the number of cards with $I$, $B$ is the number of cards with $+$, $C$ is the number of cards with $\times$, and $K$ is how many sums you must report.

Print $K$ lines. On line $i$, print the sum, taken over every possible row of cards, of the number that sits $i$-th from the top of the stack after the procedure ends, modulo $1\,000\,000\,007$.