[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 collections import deque
2   MOD = 10 ** 9 + 7
3   def solve(n, edge_list):
4       factorial = [1] * n
5       for i in range(1, n):
6           factorial[i] = (factorial[i - 1] * i) % MOD
7       factorial_inv = [1] * n
8       factorial_inv[-1] = pow(factorial[-1], MOD - 2, MOD)
9       for i in range(n - 2, 0, -1):
10          factorial_inv[i] = (factorial_inv[i + 1] * (i + 1)) % MOD
11      g = [[] for _ in range(n)]
12      for i in range(n - 1):
13          a, b = edge_list[i]
14          g[a].append(b)
15          g[b].append(a)
16      order = []
17      parent = [-1] * n
18      parent[0] = 0
19      children = [[] for _ in range(n)]
20      queue = deque([0])
21      while len(queue) > 0:
22          p = queue.popleft()
23          order.append(p)
24          for q in g[p]:
25              if parent[q] == -1:
26                  parent[q] = p
27                  children[p].append(q)
28                  queue.append(q)
29      count_children_res = [-1] * n
30      for x in order[::-1]:
31          if len(children[x]) == 0:
32              count_children_res[x] = 0
33          else:
34              total = 0
35              for y in children[x]:
36                  total += count_children_res[y]
37              r = len(children[x]) + total
38              count_children_res[x] = r
39      assign_children_res = [-1] * n
40      for x in order[::-1]:
41          if len(children[x]) == 0:
42              assign_children_res[x] = 1
43          else:
44              r = factorial[count_children_res[x]]
45              for y in children[x]:
46                  r *= assign_children_res[y]
47                  r *= factorial_inv[count_children_res[y] + 1]
48                  r %= MOD
49              assign_children_res[x] = r
50      assign_parents_res = [-1] * n
51      assign_parents_res[0] = 1
52      for x in order[1:]:
53          p = parent[x]
54          r = assign_children_res[p]
55          r *= pow(assign_children_res[x], MOD - 2, MOD)
56          r *= factorial[count_children_res[x] + 1]
57          r *= factorial_inv[count_children_res[p]]
58          r *= assign_parents_res[p]
59          r *= factorial[n - count_children_res[x] - 2]
60          r *= factorial_inv[n - count_children_res[p] - 1]
61          r %= MOD
62          assign_parents_res[x] = r
63      res_list = [0] * n
64      res_list[0] = assign_children_res[0]
65      for i in range(1, n):
66          res = factorial[n - 1]
67          for j in children[i]:
68              res *= assign_children_res[j]
69              res *= factorial_inv[count_children_res[j] + 1]
70              res %= MOD
71          res *= assign_parents_res[i]
72          res *= factorial_inv[n - 1 - count_children_res[i]]
73          res %= MOD
74          res_list[i] = res
75      return res_list
76  def main():
77      n = int(input())
78      edge_list = []
79      for _ in range(n - 1):
80          a, b = map(int, input().split())
81          edge_list.append((a - 1, b - 1))
82      res_list = solve(n, edge_list)
83      for res in res_list:
84          print(res)
85  def test():
86      assert solve(3, [(0, 1), (0, 2)]) == [2, 1, 1]
87      assert solve(2, [(0, 1)]) == [1, 1]
88      assert solve(5, [(0, 1), (1, 2), (2, 3), (2, 4)]) == [2, 8, 12, 3, 3]
89      assert solve(8, [(0, 1), (1, 2), (2, 3), (2, 4), (2, 5), (5, 6), (5, 7)]) == [40, 280, 840, 120, 120, 504, 72, 72]
90  if __name__ == "__main__":
91      test()
92      main()
```


**Question**: Does line `41` have control dependence over line `53` in function `solve`? If so, provide a trace.

**Output**:


Ground truth answer: no