[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   import sys, bisect, math, itertools, string, queue, copy
2   from collections import Counter,defaultdict,deque
3   from itertools import permutations, combinations
4   from heapq import heappop, heappush
5   input = sys.stdin.readline
6   sys.setrecursionlimit(10**8)
7   mod = 10**9+7
8   def inp(): return int(input())
9   def inpm(): return map(int,input().split())
10  def inpl(): return list(map(int, input().split()))
11  def inpls(): return list(input().split())
12  def inplm(n): return list(int(input()) for _ in range(n))
13  def inplL(n): return [list(input()) for _ in range(n)]
14  def inplT(n): return [tuple(input()) for _ in range(n)]
15  def inpll(n): return [list(map(int, input().split())) for _ in range(n)]
16  def inplls(n): return sorted([list(map(int, input().split())) for _ in range(n)])
17  def bfs(n,g):
18      went = [False]*n
19      dis = [0 for _ in range(n)]
20      que = deque([])
21      que.append(0)
22      went[0] = True
23      cnt_odd = 0
24      cnt_even = 1
25      while que:
26          go = que.pop()
27          for i in range(len(g[go])):
28              if went[g[go][i]]:
29                  continue
30              dis[g[go][i]] = dis[go] + 1
31              went[g[go][i]] = True
32              if dis[g[go][i]]%2 == 0:
33                  cnt_even += 1
34              else:
35                  cnt_odd += 1
36              que.append(g[go][i])
37      return dis,cnt_odd,cnt_even
38  def length3(s,n,g,ans,que_1,que_2):
39      went = [False]*n
40      dis = [0] * n
41      see = []
42      que = deque([])
43      que.append(s)
44      went[s] = True
45      while que:
46          go = que.pop()
47          for i in range(len(g[go])):
48              if went[g[go][i]]:
49                  continue
50              dis[g[go][i]] = dis[go] + 1
51              went[g[go][i]] = True
52              if dis[g[go][i]] < 3:
53                  que.append(g[go][i])
54              elif dis[g[go][i]] == 3:
55                  see.append(g[go][i])
56      check1 = 0
57      check2 = 0
58      for e in see:
59          if ans[e] == 0:
60              continue
61          if ans[e]%3 == 1:
62              check1 += 1
63          if ans[e]%3 == 2:
64              check2 += 1
65      if check1>0 and check2>0:
66          print(-1)
67          exit()
68      if check1>0:
69          if que_2:
70              ans[s] = que_2.pop()
71          else:
72              print(-1)
73              exit()
74          for e in see:
75              if ans[e] == 0:
76                  if que_1:
77                      ans[e] = que_1.pop()
78                  else:
79                      print(-1)
80                      exit()
81      elif check2>0:
82          if que_1:
83              ans[s] = que_1.pop()
84          else:
85              print(-1)
86              exit()
87          for e in see:
88              if ans[e] == 0:
89                  if que_2:
90                      ans[e] = que_2.pop()
91                  else:
92                      print(-1)
93                      exit()
94      else:
95          if len(que_1) > len(que_2):
96              if que_2:
97                  ans[s] = que_2.pop()
98              else:
99                  print(-1)
100                  exit()
101              for e in see:
102                  if ans[e] == 0:
103                      if que_1:
104                          ans[e] = que_1.pop()
105                      else:
106                          print(-1)
107                          exit()
108          else:
109              if que_1:
110                  ans[s] = que_1.pop()
111              else:
112                  print(-1)
113                  exit()
114              for e in see:
115                  if ans[e] == 0:
116                      if que_2:
117                          ans[e] = que_2.pop()
118                      else:
119                          print(-1)
120                          exit()
121      return ans,que_1,que_2
122  def main():
123      n = inp()
124      g = [[] for _ in range(n)]
125      for _ in range(n-1):
126          a,b = inpm()
127          g[a-1].append(b-1)
128          g[b-1].append(a-1)
129      dis,cnt_odd,cnt_even = bfs(n,g)
130      ans = [0 for _ in range(n)]
131      if min(cnt_even,cnt_odd) <= n//3:
132          check = [1 for _ in range(n)]
133          x = 1
134          if cnt_even <= cnt_odd:
135              for i in range(n):
136                  if dis[i]%2 == 0:
137                      ans[i] = 3*x
138                      check[3*x-1] = 0
139                      x += 1
140              x = 0
141              for i in range(n):
142                  if ans[i] == 0:
143                      while not check[x]:
144                          x += 1
145                      ans[i] = x + 1
146                      x += 1
147              print(*ans)
148          elif cnt_even >= cnt_odd:
149              for i in range(n):
150                  if dis[i]%2 == 1:
151                      ans[i] = 3*x
152                      check[3*x-1] = 0
153                      x += 1
154              x = 0
155              for i in range(n):
156                  if ans[i] == 0:
157                      while not check[x]:
158                          x += 1
159                      ans[i] = x + 1
160                      x += 1
161              print(*ans)
162          return
163      check = [1 for _ in range(n)]
164      x = 1
165      if cnt_even <= cnt_odd:
166          remain = []
167          for i in range(n):
168              if dis[i]%2 == 0:
169                  if x*3 <= n:
170                      ans[i] = 3*x
171                      check[3*x-1] = 0
172                      x += 1
173                  else:
174                      remain.append(i)
175          que_1 = deque()
176          que_2 = deque()
177          for i in range(n):
178              if (i+1)%3 == 1:
179                  que_1.append(i+1)
180              elif (i+1)%3 == 2:
181                  que_2.append(i+1)
182          for e in remain:
183              ans,que_1,que_2 = length3(e,n,g,ans,que_1,que_2)
184          for i in range(n):
185              if ans[i] == 0:
186                  if que_1:
187                      ans[i] = que_1.pop()
188                  else:
189                      ans[i] = que_2.pop()
190          print(*ans)
191      else:
192          remain = []
193          for i in range(n):
194              if dis[i]%2 == 1:
195                  if x*3 <= n:
196                      ans[i] = 3*x
197                      check[3*x-1] = 0
198                      x += 1
199                  else:
200                      remain.append(i)
201          que_1 = deque()
202          que_2 = deque()
203          for i in range(n):
204              if (i+1)%3 == 1:
205                  que_1.append(i+1)
206              elif (i+1)%3 == 2:
207                  que_2.append(i+1)
208          for e in remain:
209              ans,que_1,que_2 = length3(e,n,g,ans,que_1,que_2)
210          for i in range(n):
211              if ans[i] == 0:
212                  if que_1:
213                      ans[i] = que_1.pop()
214                  else:
215                      ans[i] = que_2.pop()
216          print(*ans)
217  if __name__ == "__main__":
218      main()
```


**Question**: Does line `45` have control dependence over line `53` in function `length3`? If so, provide a trace.

**Output**:


Ground truth answer: yes