Solve the programming task below in a Python markdown code block.
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You are given a graph $G$ with $N$ vertices (numbered $1$ through $N$) and $M$ edges. You should partition the vertices of $G$ into two sets $A$ and $B$ such that:
each vertex of $G$ belongs to exactly one of these sets
$A$ is non-empty
$A$ is an independent set in $G$, i.e. for each pair of vertices $u, v \in A$, $G$ does not contain an edge $(u, v)$
for each vertex $a \in A$ and each vertex $b \in B$, there is an edge $(a, b)$ in $G$

Find the number of such partitions $(A, B)$. Also, give an example of one of these partitions or determine that no such partition exists.

Two partitions are considered different if there is a vertex that is in the set $A$ in one partition and in the set $B$ in the other partition.

------  Input ------
The first line of the input contains a single integer $T$ denoting the number of test cases. The description of $T$ test cases follows.
The first line of each test case contains two space-separated integers $N$ and $M$.
Each of the next $M$ lines contains two space-separated integers $u$ and $v$ denoting that vertices $u$ and $v$ are connected by an edge.

------  Output ------
For each test case, print two lines.
The first of these lines should contain a single integer — the number of partitions satisfying the given conditions.
The second line should contain a single string with length $N$. If there are no partitions satisfying the given conditions, each character of this string should be '0'. Otherwise, this string should describe one such partition — for each valid $i$, the $i$-th character should be '1' if vertex $i$ belongs to $A$ or '0' if it belongs to $B$.

------  Constraints  ------
$1 ≤ T ≤ 5 \cdot 10^{5}$
$1 ≤ N, M ≤ 5 \cdot 10^{5}$
$1 ≤ u, v ≤ N$
there are no self-loops or duplicate edges in the graph
the sum of $N$ over all test cases does not exceed $5 \cdot 10^{5}$
the sum of $M$ over all test cases does not exceed $5 \cdot 10^{5}$

------  Subtasks ------
Subtask #1 (30 points): $N,M,T ≤ 100$

Subtask #2 (70 points): original constraints

----- Sample Input 1 ------ 
1

5 7

1 3

1 4

1 5

2 3

2 4

2 5

3 4
----- Sample Output 1 ------ 
1

11000
----- explanation 1 ------ 
Example case 1: The only possible partition satisfying the given conditions is $A = \{1, 2\}, B = \{3, 4, 5\}$.