Followers of I 3

Hyeonjong joined the church of $I$, which holds the number $I$ sacred. The church treats $I$ itself, and every other number that arithmetic on $I$ can produce, as a good number. To make many good numbers, Hyeonjong prepared the following game.

He needs two things.

• $A$ cards with 'I' drawn on both sides, $B$ cards with '+', and $C$ cards with '×'.
• A stack that already holds infinitely many copies of $I$.

Hyeonjong shuffles the cards, draws them one at a time in the shuffled order, and acts on each card he draws.

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

The stack always holds infinitely many numbers, so every draw order runs to the end.

Once every card is drawn, the number on top of the stack is a good number, and so is every number below it. Hyeonjong therefore wants, over all possible card arrangements, the sum of the topmost numbers, the sum of the second numbers from the top, and so on down to the sum of the $K$-th numbers from the top.

Cards drawn with the same symbol are not distinguished. An arrangement is one distinct sequence of $A+B+C$ symbols, and each arrangement is counted exactly once.

The first line contains five integers $I$, $A$, $B$, $C$, and $K$, separated by spaces. $I$ is the sacred number, $A$ is the number of cards with 'I', $B$ is the number of cards with '+', $C$ is the number of cards with '×', and $K$ is how many sums to report.

• $1 \le I \le 10^9$
• $0 \le A, B, C \le 40$
• $1 \le K \le 40$

Print $K$ lines. On line $i$, print the sum, over all possible card arrangements, of the number that sits $i$-th from the top of the stack once every card is drawn. The sum can be very large, so print it modulo $1\,000\,000\,007$.