[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 "Is there information flow from `(varA,lineA)` to `(varB,lineB)`? If so, provide one feasible trace.", respond with a JSON object:

- If information flow exists:
```json
{
  "InformationFlow": true,
  "Trace": [
    {
      "from": ["varOrExpr", lineNumber            /*, "use" optional */],
      "to":   ["varOrExpr", lineNumber            /*, "use" optional */],
      "type": "data" | "control"
    },
    // … additional edges
  ]
}
```
- If no information flow exists, omit the Trace field:
```json
{
  "InformationFlow": false
}
```

- Every trace edge must specify "from", "to" (each may include a third "use" element when the line is a conditional statement that only reads without redefining/updating the value), and "type".
- **We do not consider loop in the trace**.


A **trace** represents a **transitive chain of direct flows** arise from compositions of **direct** implicit or explicit flows, where each edge represents a **direct implicit flow** (through control dependence) or a **direct explicit flow** edge (through data dependence).


You only need to provide **one** valid trace if you conclude a information flow exists, even if multiple possible chains exist. 


## 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:
Is there information flow from `(balance,4)` to `(transaction,10)`? If so, provide a trace.

**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)`. Here we focus on `(status,7)`. 
- 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.
  
Hence, A transitive of flow of direct implicit or explicit flows forms a information flow trace.


**Output**:
```json
{
  "InformationFlow": true,
  "Trace": [
    {
      "from": ["balance", 4],
      "to":   ["balance", 5],
      "type": "data"
    },
    {
      "from": ["balance", 5],
      "to":   ["balance", 6, "use"],
      "type": "data"
    },
    {
      "from": ["balance", 6, "use"],
      "to":   ["status", 7],
      "type": "control"
    },
    {
      "from": ["status", 7],
      "to":   ["limit", 9],
      "type": "data"
    },
    {
      "from": ["limit", 9],
      "to":   ["transaction", 10],
      "type": "data"
    }
  ]
}
```

 
- The edge from line 6 to `(status,7)` is through control dependence and therefore with a type of "control". 
- Line 6 only reads the value of `balance` without defining/updating it. So it is marked with `"use"`. 
- You only need to provide **one** valid chain if you conclude a the information flow exists. 


#### Example Question 1.2

Is there information flow from  `(flag,3)` to `(limit,9)`? If so, provide a trace.

**Analysis**
- Line 9: `(limit,9)` reads `status`, so there is a direct explicit flow from either `(status,7)` (if line 6’s condiation evaluates to `true`) or on the original `(status,2)` (if not) to `(limit,9)`. 
- Since this is a static analysis, all possible paths are considered without analyzing actual reachability. This means line 6 is treated as being evaluated to both possibly true and possibly false.
  - If line 6 is true, `(limit,9)` depends on `(status,7)`. 
    - Line 7: `(status,7)` doesn't have any data dependence as it is assigned with a constant number. The execution of line 7 is control-dependent on the condition in line 6.
    - Line 6: the condition at line 6 directly reads from `(balance,5)`. 
    - Line 5: `(balance,5)` is updated based on the previous value of `balance` in line 4.
    - Line 4: `(balance,4)` is initialized with a constant number. Therefore it has no more data dependence. Nor is it guarded by any conditions.
  - If line 6 is not true, `(limit,9)` depends on `(status,2)`. 
    - Line 2: `(status,2)` is initialized with a constant number. Therefore it has no more data dependence. Nor is it guarded by any conditions.
- Therefore, among all the possible paths, there is no transitive (indirect) chain of implicit or explicit flow from `(flag,3)` to `(limit,9)`. The information flow doesn't exist.


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

#### Example Question 1.3

Is there information flow from  `(status,2)` to `(transaction,10)`? If so, provide a trace.

**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`, o there is a direct explicit flow from either `(status,7)`, or the original `(status,2)`, to `(limit,9)`.
- Similar to the above example, line 6 is possibly evaluated to true or false.
- If line 6 evaluates to `true`, there is a direct explicit flow from `(status,7)` to `(limit,9)`, which directly data-dependent on `(status,2)`, forming a transitive explicit flow trace :`(status,2) -> (status,7) -> (limit,9) -> (transaction,10)`.
- If line 6 evaluates to `false`, `(limit,9)` has a direct data dependence on `(status,2)`, forming a transitive explicit flow trace :`(status,2) -> (limit,9) -> (transaction,10)`.
- A transitive of explicit flow forms a information flow trace. Only one valid trace is required for the output even if multiple exists.

**Output**:
```json
{
  "InformationFlow": true,
  "Trace": [
    {
      "from": ["status", 2],
      "to":   ["limit", 9],
      "type": "data"
    },
    {
      "from": ["limit", 9],
      "to":   ["transaction", 10],
      "type": "data"
    }
  ]
}
```


### 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:
Is there information flow from  `(size,2)` to `(summary,17)`? If so, provide a trace.

**Analysis**:
- Line 17: `(summary,17)` reads `diff` and `j`. 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 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.
- Line 9: `while i < size` reads `size` (a pure read), so has a direct explicit flow from `(size,2)`.
- Line 2: `size` is initialized with a constant number. It is not guarded with any conditions. 
- Therefore, there is an transitive trace of direct implicit/excplit flows from `(size,2)` to `(summary,17)`.

Only one valid trace is required for the output. 

**Output**:
```json
{
  "InformationFlow": true,
  "Trace": [
    {
      "from": ["size", 2],
      "to":   ["size", 9, "use"],
      "type": "data"
    },
    {
      "from": ["size", 9, "use"],
      "to":   ["j", 14],
      "type": "control"
    },
    {
      "from": ["j", 14],
      "to":   ["summary", 17],
      "type": "data"
    }
  ]
}
```


#### Example Question 2.2:
Is there information flow from  `(j,5)` to `(score,11)`? If so, provide a trace. 

**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)`.
- 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).
- If line 11 reads `j` from line 5, it forms a tansitive explicit flow trace: `(j,5) -> (score,11)`.
- If line 11 reads `j` from line 14, it forms a tansitive explicit flow trace: `(j,5) -> (j,14) -> (score,11)`.
- A transitive of explicit flow forms a information flow trace. Only one valid trace is required for the output even if multiple exists.


**Output**:
```json
{
  "InformationFlow": true,
  "Trace": [
    {
      "from": ["j", 5],
      "to":   ["score", 11],
      "type": "data"
    }
  ]
}
```



#### Example Question 2.3:
Is there information flow from  `(j,1)` to `(i,15)`? If so, provide a trace.

**Analysis**: 
- Line 15: `(i,15)` updates `i` based on its previous value. So there is a direct explicit flow from either `(i,4)` (initialization) or itself (loop-carried) to `(i,15)`. But we do not consider loop-carried dependencies across iterations in this analysis. So we focus on `(i,4)`.
- The execution of line 15 is conditional on the while loop at line 9.
- Line 9: `while i < size` depends on `i` and `size`. It directly reads from `(i,4)` (or possibly `(i,15)`) and `(size,2)`.
- Both `(i,4)` and `(size,2)` are initialized with constants and are not themselves control-dependent on any earlier values or branches.
- There is no use of `(j,1)` in the computation of `(i,15)`, nor does `(j,1)` influence whether line 15 executes.
- Therefore, there is no transitive flow, explicit or implicit, from `(j,1)` to `(i,15)`.


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


#### Example Question 2.4:
Is there information flow from  `(val,1)` to `(last,16)`? If so, provide a trace.

**Analysis**:
- Line 16: `(last,16)` reads `arr[-1]`. That element may have been modified by the assignment in line 8 (or by the loop updates in line 10). Under static analysis, multiple update sites may contribute. We conservatively pick one valid trace (line 8).
- 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.
- Therefore, there exists a trace from `(val,1)` to `(last,16)`, and the edge from line 7 to `(arr,8)` is through implicit and explicit flows. 

**Output**:
```json
{
  "InformationFlow": true,
  "Trace": [
    {
      "from": ["val", 1],
      "to":   ["val", 7, "use"],
      "type": "data"
    },
    {
      "from": ["val", 7, "use"],
      "to":   ["arr", 8],
      "type": "control"
    },
    {
      "from": ["arr", 8],
      "to":   ["last", 16],
      "type": "data"
    }
  ]
}
```

---

[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**: Is there information flow from `(check1,56)` to `(e,58)` in function `length3`? If so, provide a trace.

**Output**:


Ground truth answer: no