[INSTRUCTIONS]

You are a program-analysis assistant. Your task is to statically analyze the control dependence of a given code snippet. 

## 1. Control Dependence Definition
  Control dependence captures the influence of control-flow decisions on the execution of statements.

  A statement `S2` is control-dependent on `S1` if there is a **transitive** (indirect) chain of **direct** control dependencies from `S1` to `S2`. This is equivalent to saying that `S1` has control dependence over `S2`, meaning `S1`’s condition influences whether `S2` executes.


  **Direct Control Dependence**:
  A statement `S2` is **directly** control-dependent on a statement `S1` if:
  
  1.	`S1` is a conditional control statement (e.g., an `if`, `while`, `for`, `switch`, etc.).
  
  2.	`S1` directly determines whether `S2` executes. That is, `S1` has multiple successor branches, 

  - there exists **at least one branch** in which `S2` **always executes**, and
  - there exists **at least one other branch** in which `S2` does **not necessarily execute**.

    “Not necessarily execute” means that `S2` might execute or might not, but it is **not guaranteed** to execute in that branch.

  `S2` is **control-dependent** on `S1` if there exists a **transitive chain** of control dependencies from `S1` to `S2`, where each intermediate step in the chain represents a **direct** control dependence between two statements.



## 2. Output Format
When asked, “Does line `S1` have control dependence over line `S2`? If so, provide a trace.” You should respond in JSON format as follows:

- If there is a control dependence:

```json
{
  "ControlDependence": true,
  "Trace": [S1, ..., S2]
}
```

- If there is no control dependence:
```json
{
  "ControlDependence": false
}
```

A **trace** represents a **transitive control flow**, where each adjacent pair in the list reflects a **direct** control dependence. The trace must starts with `S1` and ends with `S2`.


    

## 3. Interprocedural Control Dependence
  All dependence analysis is performed within **a single function**. We do not track dependencies across function boundaries. The analysis only applies to variables and control structures inside the **specified function**.


## 4. Example Code Snippet

### Example 1

```python
1  if x > 0:
2      y = 10       # this line is directly control-dependent on line 1 (x>0)
3      if y > 5:
4          z = 20   # this line is directly control-dependent on line 3 (y>5)
5          w = 30   # this line is directly control-dependent on line 3 (y>5)
6  v = 40           # this line is NOT control-dependent on any line
```


**Analysis**:
- Line 2 is directly control-dependent on line 1 (`x>0`)
- Lines 4 and 5 are directly control-dependent on line 3 (`y>5`).
- Lines 4 and 5 are **indirectly** control-dependent on line 1 (`x>0`) becasue even if line 1 (`x>0`) is `true`, lines 4 and 5 may not execute if line 3’s condition (`y>5`) is `false`. However, since control dependence is transitive, lines 4 and 5 are **indirectly** control-dependent on line 1. 
- Line 6 is not control-dependent on any lines because Line 6 always executes, regardless of whether line 1 (`x>0`) or line 3 (`y>5`) is true or false.



#### Example Question 1.1: 
Does line `1` have control dependence over line `5`? If so, provide a trace.

**Output**: 
```json
{
  "ControlDependence": true,
  "Trace": [1, 3, 5]
}
```

#### Example Question 1.2:
Does line `3` have control dependence over line `6`? If so, provide a trace.

**Output**: 
```json
{
  "ControlDependence": false
}
```


### Example 2

```python
1  count = 0
2  if count < 5:
3      count += 1
4  print("Step 1 done")
5  while count < 10:
6      if count == 7:
7          break
8      if count % 2 == 0:
9          continue
10     count += 2
11     if count > 9:
12         count = 9
13     print("Iteration done")
14 print("End of program")
```

#### Example Question 2.1: 
Does line `5` have control dependence over line `13`? If so, provide a trace.

**Analysis**:
  - Line 13 is directly control-dependent on line 8 because if line 8 evaluates to `true`, continue skips line 13 for that iteration.
  - Line 8 is directly control-dependent on line 6 because if line 6 evaluates to `true`, the break statement terminates the loop, preventing line 8 from executing.
  - Line 6 is directly control-dependent on line 5 because the loop condition at line 5 determines whether line 6 executes. If line 5 evaluates to `false`, execution skips the loop entirely.


Since control dependence is transitive, line 5 indirectly controls line 13 through lines 6 and 8. The trace is `5->6->8->13`.

**Output**:
```json
{
  "ControlDependence": true,
  "Trace": [5, 6, 8, 13]
}
```

#### Example Question 2.2: 
Does line `8` have control dependence over line `10`? If so, provide a trace.


**Analysis**: Similar to the explanation above, line 8 is an `if` condition inside the `while` loop with a `continue`, so it can make the execution skip the rest of the loop entirely, including line 10. Therefore, line 10 directly control-dependent on 8.



**Output**:
```json
{
  "ControlDependence": true,
  "Trace": [8, 10]
}
```

#### Example Question 2.3: 
Does line `5` have control dependence over line `14`? If so, provide a trace.

**Analysis**: Line 5 doesn't affect whether line 14 is reached. Line 14 is outside the while loop (lines 5 -- 13) and will be executed regardless of the condition. 

**Output**:
```json
{
  "ControlDependence": false
}
```


#### Example Question 2.4: 
Does line `2` have control dependence over line `12`? If so, provide a trace.

**Analysis**: Line 2 doesn't affect whether line 12 is reached. The condition in line 2 only controls whether line 3 is executed. Line 12 is in while loop (line 5 -- 13) and will or will not be executed regardless of the condition in line 2. 

**Output**:
```json
{
  "ControlDependence": false
}
```

---

[YOUR TURN]

Below is **your target snippet**. 

```Python
1   from sys import stdin
2   from itertools import repeat
3   def main():
4       n, m = map(int, stdin.readline().split())
5       dat = map(int, stdin.read().split())
6       for _  in range(2 * m):
7           dat.append(10)
8       la = [] 
9       for _ in range(n + 1): 
10          la.append(None) 
11      xt = [] 
12      for _ in range(dat): 
13          xt.append(None) 
14      j = 0
15      for i in xrange(m):
16          x, y = dat[j:j+2]
17          xt[j] = la[y]
18          xt[j+1] = la[x]
19          la[y] = j
20          la[x] = j + 1
21          j += 2
22      done = [] 
23      for _ in range(n + 1): 
24          done.append(None)  
25      x = 1
26      po = []
27      while 1:
28          po.append(x)
29          done[x] = 1
30          y = la[x]
31          while y is not None:
32              z = dat[y]
33              if done[z] is None:
34                  x = z
35                  break
36              y = xt[y]
37          else:
38              break
39      pa = []
40      while 1:
41          pa.append(x)
42          done[x] = 1
43          y = la[x]
44          while y is not None:
45              z = dat[y]
46              if done[z] is None:
47                  x = z
48                  break
49              y = xt[y]
50          else:
51              break
52      po = po[::-1] + pa[1:]
53      print len(po)
54      print ' '.join(map(str, po))
55  main()
56  from sys import stdin
57  from itertools import repeat
58  def main():
59      n, m = map(int, stdin.readline().split())
60      dat = map(int, stdin.read().split(), repeat(10, 2 * m))
61      la = [None] * (n + 1)
62      xt = [None] * len(dat)
63      j = 0
64      for i in xrange(m):
65          x, y = dat[j:j+2]
66          xt[j] = la[y]
67          xt[j+1] = la[x]
68          la[y] = j
69          la[x] = j + 1
70          j += 2
71      done = [None] * (n + 1)
72      x = 1
73      po = []
74      while 1:
75          po.append(x)
76          done[x] = 1
77          y = la[x]
78          while y is not None:
79              z = dat[y]
80              if done[z] is None:
81                  x = z
82                  break
83              y = xt[y]
84          else:
85              break
86      pa = []
87      while 1:
88          pa.append(x)
89          done[x] = 1
90          y = la[x]
91          while y is not None:
92              z = dat[y]
93              if done[z] is None:
94                  x = z
95                  break
96              y = xt[y]
97          else:
98              break
99      po = po[::-1] + pa[1:]
100      print len(po)
101      print ' '.join(map(str, po))
102  main()
```


**Question**: Does line `23` have control dependence over line `34` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: no