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Ex - Fill Triangle

Score : $600$ points

Problem Statement

Blocks are stacked in a triangle. The $i$-th column from the top has $i$ blocks.

You are given a sequence $P = ((a_1, c_1), (a_2, c_2), ..., (a_M, c_M))$ which is a result of the run-length compression of a sequence $A = (A_1, A_2, ..., A_N)$ consisting of non-negative integers less than or equal to $6$.

• For example, when $A = (2, 2, 2, 5, 5, 1)$, you are given $P = ((2, 3), (5, 2), (1, 1))$.

You will write a number on each block so that the following conditions are satisfied, where $B_{i,j}$ denotes the number to write on the $j$-th block from the left in the $i$-th column from the top:

• For all integers $i$ such that $1 \leq i \leq N$, it holds that $B_{N,i} = A_{i}$.
• For all pairs of integers $(i, j)$ such that $1 \leq j \leq i \leq N-1$, it holds that $B_{i,j}= (B_{i+1,j}+B_{i+1,j+1})\bmod 7$.

Enumerate the numbers written on the blocks in the $K$-th column from the top.

What is run-length compression?

The run-length compression is a conversion from a given sequence $A$ to a sequence of pairs of integers obtained by the following procedure.

1. Split $A$ off at the positions where two different elements are adjacent to each other.
2. For each subsequence $B$ that has been split off, replace $B$ with a integer pair of "the number which $B$ consists of" and "the length of $B$".
3. Construct a sequence consisting of the integer pairs after the replacement without changing the order.

Constraints

• $1 \leq N \leq 10^9$
• $1 \leq M \leq \min(N, 200)$
• $1 \leq K \leq \min(N,5 \times 10^5)$
• $0 \leq a_i \leq 6$
• $1 \leq c_i \leq N$
• $\sum_{i=1}^M c_i = N$
• All values in input are integers.

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Input

Input is given from Standard Input in the following format:

$N$ $M$ $K$

$a_1$ $c_1$

$a_2$ $c_2$

$\vdots$

$a_M$ $c_M$

Output

Print the answer in the following format. It is guaranteed that the answer is unique under the Constraint of the problem.

``` $B_{K,1}$ $B_{K,2}$ $\dots$ $B_{K,K}$ ```

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6 3 4
2 3
5 2
1 1

1 4 3 2

We have $A = (2,2,2,5,5,1)$. The number written on the blocks are as follows.

``` 3 5 5 5 0 5 1 4 3 2 4 4 0 3 6 2 2 2 5 5 1 ```

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1 1 1
6 1

6

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111111111 9 9
0 1
1 10
2 100
3 1000
4 10000
5 100000
6 1000000
0 10000000
1 100000000

1 0 4 2 5 5 5 6 3