[INSTRUCTIONS]

You are a program-analysis assistant. Please perform a **static** analysis of “information flow” on a given code snippet, treating each branch or loop condition as potentially taking any outcome, without using semantic or symbolic execution to prune paths.

## 1. Information Flow Definition

An **information flow** from variable `x` to variable `y`, written `x -> y`, exists whenever information stored in `x` is transferred to or used to derive information transferred to `y`. 

We denote each variable instance as `(var,lineNumber)`, meaning the variable `var` is **defined or updated** at `lineNumber`.


### Types of Direct Flows

- **Direct Explicit Flow**  
  A direct explicit flow occurs when the value written by one variable instance is directly used in computing the value of another, through operations like assignments.

  Formally, if `(varA, lineA)` writes a value and `(varB, lineB)` reads that value to compute its own, then there is a **direct explicit flow** from `(varA, lineA)` to `(varB, lineB)`, written as: `(varA,lineA) -> (varB,lineB)`. 


  **Example 1**:
    ```
    1  x = 5
    2  y = x + 1
    ```
    `(x,1) -> (y,2)` is a direct explicit flow.


  **Example 2**:
    ```
    1  x = 5
    2  if cond:
    3      y = x + 1
    ```
    `(x,1) -> (y,3)` is a direct explicit flow. 

    Under static analysis, we treat **each conditional or loop as potentially taking any outcome**, and we **do not perform semantic pruning**. Therefore, even if the statement at line 3 is conditionally executed, we still recognize `(x,1)` as used to compute `(y,3)`.
    


- **Direct Implicit Flow**

  A direct implicit flow from `(varA,lineA)` to `(varB, lineB)` occurs when `(varA,lineA)`’s value directly influences whether `lineB` executes. In other words, it corresponds to a **direct control dependence**, where the condition has **at least one branch** where `(varB, lineB)` must run and another where it may not. All conditional structures (e.g., if, while, for, switch, etc.) generate these flows.

  Formally, if `(varA,lineA)` is **directly read** in a condition at `lineC` and that condition **directly** decides whether `lineB` executes, then we say there is an **direct implicit flow** from `(varA,lineA)` to `(varB,lineB)` (or `(varA,lineA) -> (varB,lineB)`).

  **Example**:
  ```
  1  z = x + 5
  2  if z > 1:
  3      y = 10
  ```
  `(z,1) -> (z,2)* -> (y,3)` is a direct implicit flow through the condition at line 2. Line 2 is a conditional statement and the value of `(z,2)*` here is not redefined, therefore marked with "*".
  
  The asterisk * indicates that the variable is used for the conditional statement but not redefined at that line (e.g., used in a condition).



### Information Flow Between Variables

An **information flow** exists from a variable `(varA, lineA)` to another variable `(varB, lineB)` if there is a **transitive chain of direct flows** — explicit, implicit, or both — connecting them.

That is, there exists a sequence of variables:
`(varA, lineA) -> ... -> (varB, lineB)`



## 2. **Output Format**  

When asked "Which variable instances have information flow over `(targetVar, targetLine)`? List all such variables.", respond:, respond with a JSON object:

```json
   {
    "InfomationFlowSources": [
      ["valA", lineA],
      ["valB", lineB],
      ...
    ]
   }
   ```

If you believe there is no variables have information flow on `(targetVar, targetLine)`, respond:

```json
{
  "InfomationFlowSources": false
}
```


## 3. **Intraprocedural Data 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  def example_func():
2      status = 0
3      flag = False
4      balance = 1000
5      balance += 500
6      if balance > 1000:
7          status = 1
8          flag = True
9      limit = status * 5000
10     transaction = limit * 0.2
11     return flag
```

#### Example Question 1.1:
Which variable instances have information flow over `(transaction,10)`? List all such variables.


**Analysis**:
- Line 10: `(transaction,10)` reads `limit`, so there is a direct explicit flow from `(limit,9)` to `(transaction,10)`. 
- Line 9: `(limit,9)` reads `status`, so there is a direct explicit flow from `(status,7)` (or `(status,2)`) to `(limit,9)`. 
- Line 7: `(status,7)` is updated only if line 6’s condition is true, so has a direct control dependence on line 6.
- Line 6: checks `balance > 1000`, reads from `(balance,5)`, so there is a direct implicit flow from `(balance,5)` to `(status,7)` through `balance` at line 6.
- Line 5: `(balance,5)` reads from `(balance,4)`, so there is a direct explicit flow from `(balance,4)` to `(balance,5)`.
- Line 4: `(balance,4)` initializes the variable balance.
  

**Output**:
```json
   {
    "InfomationFlowSources": [
      ["limit", 9],
      ["status", 2],
      ["status", 7],
      ["balance", 4],
      ["balance", 5],
    ]
   }
```

 
#### Example Question 1.2
Which variable instances have information flow over `(flag,3)`? List all such variables.

**Analysis**
Line 3 initializes `flag` without any data dependency or control dependency. 


**Output**:
```json
{
  "InfomationFlowSources": []
}
```



### Example 2
```python
1.  val = 5
2.  size = 3
3.  arr = [0] * size
4.  i = 0
5.  j = 1
6.  total = 0
7.  if val > 2:
8.      arr[j % size] = j
9.  while i < size:
10.     arr[j] += 1
11.     score = arr[j+1]
12.     total += score * 2
13.     diff = total - score
14.     j = (j + 1) % size
15.     i += 1
16. last = arr[-1]
17. summary = diff + j
```

#### Example Question 2.1:
Which variable instances have information flow over `(summary,17)`? List all such variables.

**Output**:
```json
   {
    "InfomationFlowSources": [
      ["diff", 13],
      ["j", 14],
      ["j", 5],
      ["size", 2],
      ["i", 4],
      ["i", 15],
    ]
   }
```


**Analysis**:
- Line 17: `(summary,17)` reads `diff` and `j`. `diff` is from line 13. Next, we focus on the path through `j`. `j` is updated from line 14 and line 5. 
- Static analysis assumption: Every branch or loop condition may succeed or fail; we do not prune unreachable paths. When line 9 never evaluates to `true`, `(summary,17)` reads `j` from `(j,5)`. When line 9 evaluates to `true`, line 14 will be executed. 
- Line 14: `(j,14)` is updated inside the `while i < size` loop, so its update is control-dependent on the loop condition at line 9. There is also an explicit flow from `size` to `j`.
- Line 9: `while i < size` reads `size` (a pure read), so has a direct explicit flow from `(size,2)`. It also reads `i` from line 15 or line 4.

**Output**:
```json
   {
    "InfomationFlowSources": [
      ["diff", 13],
      ["j", 14],
      ["j", 5],
      ["size", 2],
      ["i", 4],
      ["i", 15]
    ]
   }
```


#### Example Question 2.2:
Which variable instances have information flow over `(score,11)`? List all such variables.


**Analysis**:
- Line 11: `(score,11)` reads `arr[j+1]`. There is a direct data dependence on both `arr` and `j`. That `j` may come from either `(j,5)` (initial) or subsequent updates at `(j,14)`. `arr` may come from line 3, line 8, or line 10 given the index `j+1` is with an unknown value. 
- Static analysis assumption: Every branch or loop condition may succeed or fail; we do not prune unreachable paths. Therefore, `j` could be read from either line 5 (e.g., when it is the first iteration of the while loop) or line 14 (e.g., when it is the not the first iteration of the while loop). Similar for `arr`, it could be read from line 3, line 8, or line 10.
- If `arr` is read from line 10, there is an explicit flow from `(j,5)`, `(j,14)`, and `(size,2)` to `arr`. There is also an explicit flow from `arr` itself because the operation is `+=`, hence from `(arr,3)`, `(arr,8)`, or `(arr,10)`. There is also an implicit flow from `i` and `size` to line 10. Hence from `(i, 4)`, `(i,15)`, and `(size,2)`.
- If `arr` is read from line 8, there is an explicit flow from `(j,5)` and `(size,2)` to `arr`. There is also implicit flow from `val` (as in the condition in line 7) read from line 1.
- If `arr` is read from line 3, there is an explicit flow from `(size,2)` to `arr`.
- Line 11 is control-dependent on line 9, therefore there is a implicit flow from `i` (possibly from `(i,15)` or `(i,4)`) and `size` (possibly from `(size,2)`) to line 11.

**Output**:
```json
   {
    "InfomationFlowSources": [
      ["arr", 10],
      ["arr", 8],
      ["arr", 3],
      ["j", 14],
      ["j", 5],
      ["size", 2],
      ["i", 4],
      ["i", 15],
      ["val", 1]
    ]
   }
```



#### Example Question 2.3:
Which variable instances have information flow over `(last,16)`? List all such variables.

**Analysis**:
- Line 16: `(last,16)` reads `arr[-1]`. That element may have been modified by the assignment in line 8, by the loop updates in line 10, or by initialization at line 3. Under static analysis, multiple update sites may contribute. 
- If from Line 3: there is an explicit flow from `(size,2)` to it.
- If from line 8: `arr[j % size] = j` executes only if `val > 2`, so `(arr,8)` has a direct control dependence on `(val,1)` via the `if` at line 7. There is also an explicit flow from `(j,5)` and `(size,2)` to `arr`.
- If from line 10:  there is an explicit flow from `(j,5)`, `(j,14)`, and `(size,2)` to `arr`. There is also an explicit flow from `arr` itself because the operation is `+=`, hence from `(arr,3)`, `(arr,8)`, or `(arr,10)`. There is also an implicit flow from `i` and `size` to line 10. Hence from `(i, 4)`, `(i,15)`, and `(size,2)`.


**Output**:
```json
   {
    "InfomationFlowSources": [
      ["arr", 10],
      ["arr", 8],
      ["arr", 3],
      ["j", 14],
      ["j", 5],
      ["size", 2],
      ["i", 4],
      ["i", 15],
      ["val", 1]
    ]
   }
```

---

[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**: Which variable instances have information flow over `(check2,64)` in function `length3`? List all such variables.

**Output**:


Ground truth answer: yes