World Map

Mr. Pacha, a Bolivian archeologist, discovered an ancient document near Tiwanaku that describes the world during the Tiwanaku Period (300-1000 CE). At that time, there were $N$ countries, numbered from $1$ to $N$.

In the document, there is a list of $M$ different pairs of adjacent countries: $$(A[0], B[0]), (A[1], B[1]), \ldots, (A[M-1], B[M-1]).$$ For each $i$ ($0 \leq i < M$), the document states that country $A[i]$ was adjacent to country $B[i]$ and vice versa. Pairs of countries not listed were not adjacent.

Mr. Pacha wants to create a map of the world such that all adjacencies between countries are exactly as they were during the Tiwanaku Period. For this purpose, he first chooses a positive integer $K$. Then, he draws the map as a grid of $K \times K$ square cells, with rows numbered from $0$ to $K - 1$ (top to bottom) and columns numbered from $0$ to $K - 1$ (left to right).

He wants to color each cell of the map using one of $N$ colors. The colors are numbered from $1$ to $N$, and country $j$ ($1 \leq j \leq N$) is represented by color $j$. The coloring must satisfy all of the following conditions:

• For each $j$ ($1 \leq j \leq N$), there is at least one cell with color $j$.
• For each pair of adjacent countries $(A[i], B[i])$, there is at least one pair of adjacent cells such that one of them is colored $A[i]$ and the other is colored $B[i]$. Two cells are adjacent if they share a side.
• For each pair of adjacent cells with different colors, the countries represented by these two colors were adjacent during the Tiwanaku Period.

For example, if $N = 3$, $M = 2$ and the pairs of adjacent countries are $(1,2)$ and $(2,3)$, then the pair $(1,3)$ was not adjacent, and the following map of dimension $K = 3$ satisfies all the conditions.

In particular, a country does not need to form a connected region on the map. In the map above, country 3 forms a connected region, while countries 1 and 2 form disconnected regions.

Your task is to help Mr. Pacha choose a value of $K$ and create a map. The document guarantees that such a map exists. Since Mr. Pacha prefers smaller maps, in the last subtask your score depends on the value of $K$, and lower values of $K$ may result in a better score. However, finding the minimum possible value of $K$ is not required.

• $1 \le N \le 40$
• $0 \le M \le \frac{N \cdot (N - 1)}{2}$
• $1 \le A[i] < B[i] \le N$ for each $i$ such that $0 \le i < M$.
• The pairs $(A[0], B[0]), \ldots , (A[M - 1], B[M - 1])$ are distinct.
• There exists at least one map which satisfies all the conditions.