Graph Measurement

Andrew has a simple undirected graph on $n$ vertices ($1 \le n \le 20$). The graph has no self-loops and no multiple edges. He repeats the following measurement $k$ times. For each edge, he colors it black or white independently, with probability 50% for each color. Then he counts the number of black edges adjacent to each vertex and tells you these $n$ numbers.

Your goal is to restore the original graph from the given $k \cdot n$ numbers.

The first line contains two integers $n$ and $k$ (in all cases except the sample, $k = 800 \cdot n^2$). The next $k$ lines each contain $n$ integers. Line $i$ gives, for measurement $i$, the number of black edges adjacent to each vertex, in vertex order.

In the first line, print an integer $m$, the number of edges in the graph. In the next $m$ lines, describe each edge by the two vertices it connects. Vertices are numbered from 1 to $n$. There must be no self-loops, and at most one edge may connect any pair of vertices. The order of the edges and the order of the two vertices within each edge do not matter.

In the sample case, any valid graph is accepted, even if it is inconsistent with the input (for example, a graph with 0 edges). In all other cases, only the graph that actually generated the input is accepted.

Note that all cases except the sample have many measurements, specifically $k = 800 \cdot n^2$. There are 33 non-sample test cases.