Action Recognition Problem

The skeleton data has been widely used in computer vision tasks such as action recognition. In the skeleton model, the human body is represented by a set of body joints interconnected by bones. This naturally forms an undirected graph model: vertices are joints and edges are bones.

To incorporate the dynamics of the skeletons in a video, we may keep track of the joints across the frames and build a spatial-temporal graph for the video. The spatial-temporal graph consists of the skeleton graphs of every frame, with additional inter-frame edges connecting the same joints in two adjacent frames. Note that the skeleton graph should keep the same in all frames of the video. The following picture exhibits how a spatial-temporal graph is formed. 

Рис. 1: Formation of a spatial-temporal graph

Your task is to reverse-engineer a spatial-temporal graph. Formally, you should assign a pair of integers $(f\_v, s\_v)$ to every vertex $v$ of the graph, where $f\_v$ is the frame number and $s\_v$ is the index of the joint. Let $T, S$ denote the number of frames and the number of joints, respectively, then your labeling must simultaneously satisfy the following conditions:

• for every $1 \leq t \leq T$ and $1 \leq i \leq S$, exactly one vertex is labeled $(t, i)$;
• there is an edge between vertices labeled $(t, i)$ and $(t+1, i)$ for every $1 \leq t < T$ and $1 \leq i \leq S$; there are no inter-frame edges other than those mentioned before;
• for every $1 \leq t\_1 < t\_2 \leq T$ and $1 \leq i < j \leq S$, there is an edge between vertices labeled $(t\_1, i)$ and $(t\_1, j)$ if and only if so between $(t\_2, i)$ and $(t\_2, j)$.

The first line of the input consists of two integers $n, m$ $(1 \leq n \leq 100\\,000, 0 \leq m \leq 200\\,000)$, denoting the number of vertices and edges in the given spatial-temporal graph respectively. The vertices of the graph are indexed $1$ through $n$. 

Each of the remaining $m$ lines of the input contains two integers $u, v$ $(1 \leq u, v \leq n, u \neq v)$, denoting an undirected edge connecting vertices indexed $u$ and $v$. Each pair of vertices is connected by at most one edge. The input graph is guaranteed to be connected.

Print two integers $T, S$ in the first line of your output, denoting the number of frames and the number of joints, respectively. Then print $T$ lines, each containing $S$ integers; the $s$-th integer of the $t$-th line is the index of the vertex labeled $(t, s)$.

If multiple valid labelings exist, print the one with the maximum number of frames. If there are still multiple, any one is acceptable.