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 c++17.

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:
```cpp
<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.