[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
2   import bisect
3   def LI(): return [int(x) for x in sys.stdin.readline().split()]
4   def I(): return int(sys.stdin.readline())
5   def solve():
6       n = I()
7       p = [LI() for i in range(n)]
8       p.sort()
9       m = 1<<n
10      s = 0
11      for i,(x,y,num) in enumerate(p):
12          d = min(abs(x),abs(y))
13          s += num*d
14      ans = [s]*(n+1)
15      ans[-1] = 0
16      pow = [1<<i for i in range(n)]
17      dx = [[None]*m for i in range(n)]
18      dy = [[None]*m for i in range(n)]
19      for b in range(m):
20          lx = []
21          ly = []
22          res = b
23          for x,y,_ in p:
24              j = res&1
25              if j:
26                  lx.append(x)
27                  ly.append(y)
28              res >>= 1
29          ly.sort()
30          for j,(x,y,num) in enumerate(p):
31              i = bisect.bisect_left(lx,x)-1
32              dxj = abs(x)
33              dyj = abs(y)
34              if i >= 0:
35                  nd = x-lx[i]
36                  if nd < dxj:
37                      dxj = nd
38              i += 1
39              if i < len(lx):
40                  nd = lx[i]-x
41                  if nd < dxj:
42                      dxj = nd
43              i += 1
44              if i < len(lx):
45                  nd = lx[i]-x
46                  if nd < dxj:
47                      dxj = nd
48              i = bisect.bisect_left(ly,y)-1
49              if i >= 0:
50                  nd = y-ly[i]
51                  if nd < dyj:
52                      dyj = nd
53              i += 1
54              if i < len(ly):
55                  nd = ly[i]-y
56                  if nd < dyj:
57                      dyj = nd
58              i += 1
59              if i < len(ly):
60                  nd = ly[i]-y
61                  if nd < dyj:
62                      dyj = nd
63              dx[j][b] = dxj*num
64              dy[j][b] = dyj*num
65      m = 3**n
66      for b in range(m):
67          kx = ky = cnt = 0
68          res = b
69          for i in range(n):
70              j = res%3
71              if j:
72                  cnt += 1
73                  if j == 1:
74                      kx |= pow[i]
75                  else:
76                      ky |= pow[i]
77              res //= 3
78          s = 0
79          ans_ = ans[cnt]
80          for i in range(n):
81              d = min(dx[i][kx],dy[i][ky])
82              s += d
83              if s >= ans_:
84                  break
85          else:
86              ans[cnt] = s
87      for i in ans:
88          print(i)
89      return
90  if __name__ == "__main__":
91      solve()
```


**Question**: Does line `69` have control dependence over line `77` in function `solve`? If so, provide a trace.

**Output**:


Ground truth answer: yes