You are an expert competitive programmer. You will be given a problem statement, test case constraints and example test inputs and outputs. Please reason step by step about the solution (that must respect memory and time limits), then provide a complete implementation in python3.

Your solution must read input from standard input (cin), write output to standard output (cout).
Do not include any debug prints or additional output.

Put your final solution within a single code block:
```python
<your code here>
```

Execution time limit: 7.0 seconds
Memory limit: 256.0 MB

# Problem
You are given a bipartite graph $$$G$$$ with the vertex set in the left part $$$L$$$, in the right part $$$R$$$, and $$$m$$$ edges connecting these two sets. We know that $$$|L| = |R| = n$$$.

For any subset $$$S \subseteq L$$$, let $$$N(S)$$$ denote the set of all neighbors of vertices in $$$S$$$. We say that a subset $$$S \subseteq L$$$ in graph $$$G$$$ is tight if $$$|S| = |N(S)|$$$. We say that graph $$$G$$$ is good if $$$\forall_{S \subseteq L}, |S| \leq |N(S)|$$$.

Your task is to verify whether the graph is good and, if so, to optimize it. If the graph is not good, find a subset $$$S \subseteq L$$$ such that $$$|S| > |N(S)|$$$. However, if the graph is good, your task is to find a good bipartite graph $$$G'$$$ with the same set of vertices $$$L \cup R$$$, in which $$$\forall_{S \subseteq L}$$$, $$$S$$$ is tight in $$$G$$$ if and only if $$$S$$$ is tight in $$$G'$$$. If there are multiple such graphs, choose one with the smallest possible number of edges. If there are still multiple such graphs, print any.

## Input Format
The first line of the input contains two integers $$$n$$$ and $$$m$$$ ($$$1 \leq n \leq 10^3$$$, $$$0 \leq m \leq n^2$$$), separated by a single space. The number $$$n$$$ denotes the number of vertices in each of the sets $$$L$$$ and $$$R$$$, and the number $$$m$$$ denotes the number of edges between them.

The following $$$m$$$ lines describe the edges. Each of them contains two integers $$$l$$$ and $$$r$$$ ($$$1 \leq l \leq n$$$, $$$n+1 \leq r \leq 2 \cdot n$$$), separated by a single space, indicating that there is an edge from vertex $$$l \in L$$$ to vertex $$$r \in R$$$.

## Output Format
If the graph $$$G$$$ given in the input is not good, output one word "NO" in the first line of the output. In the second line of the output, output the number $$$k$$$, and in the third line, output $$$k$$$ numbers $$$l_1, l_2, \dots, l_k$$$, separated by single spaces. These numbers should indicate that for the set $$$S = \{l_1, l_2, \dots, l_k\}$$$, $$$|S| > |N(S)|$$$.

However, if the graph $$$G$$$ given in the input is good, output one word "YES" in the first line of the output. In the second line of the output, output the number $$$m'$$$, indicating the number of edges in the new, also good graph $$$G'$$$. Then, in the following $$$m'$$$ lines, output the edges of the graph $$$G'$$$ in the same format as given in the input.

## Examples
```input
3 8
1 4
1 5
1 6
2 4
2 5
2 6
3 5
3 6
```
```output
YES
6
1 4
1 5
2 5
2 6
3 6
3 4
```
-----
```input
3 4
1 4
1 5
2 6
3 6
```
```output
NO
2
2 3
```

## Note
In the first sample test, the graph $$$G$$$ is good; thus, we are looking for an equivalent graph with the same tight sets. The only tight set is $$$\{ 1, 2, 3 \}$$$, which remains tight in the resulting graph. Moreover, no other set is tight in the resulting graph. One can prove that no graph with less than $$$6$$$ edges and the same tight sets exists.

In the second sample test, the graph $$$G$$$ is not good. Set $$$\{ 2, 3 \}$$$ has only one neighbour — vertex $$$6$$$. Thus, $$$|\{ 2, 3 \}| > |\{ 6 \}|$$$, which is a prove that the input graph is not good.

Now solve the problem and return the code.