[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 __future__ import print_function
2   import sys
3   import math
4   in_stream = sys.stdin
5   def max_stack(num_ants, weights):
6       max_num_ants = 0
7       min_weight = list()
8       min_weight_for_1_ant = list()
9       for i in range(num_ants):
10          if(i == 0):
11              min_weight_for_1_ant.append(weights[i])
12          else:
13              min_weight_for_1_ant.append(min(min_weight_for_1_ant[i-1], weights[i]))
14          max_num_ants = 1
15      min_weight.append(min_weight_for_1_ant)
16      for num_in_stack in range(1, num_ants):
17          min_weight_for_num_ants = list()
18          for i in range(num_ants):
19              if(i + 1 < num_in_stack + 1):
20                  min_weight_for_num_ants.append(-1)
21              elif(i+1 == num_in_stack + 1):
22                  if(min_weight[num_in_stack-1][i-1] != -1 and min_weight[num_in_stack-1][i-1] <= 6 * weights[i]):
23                      min_weight_for_num_ants.append(min_weight[num_in_stack-1][i-1] + weights[i])
24                      max_num_ants = num_in_stack + 1
25                  else:
26                      min_weight_for_num_ants.append(-1)
27              else:
28                  if(min_weight[num_in_stack-1][i-1] != -1 and min_weight[num_in_stack-1][i-1] <= 6 * weights[i]):
29                      current_ant = min_weight[num_in_stack-1][i-1] + weights[i]
30                  else:
31                      current_ant = -1
32                  old_ant = min_weight_for_num_ants[i-1]
33                  if(current_ant == -1 and old_ant == -1):
34                      min_weight_for_num_ants.append(-1)
35                  elif(current_ant == -1):
36                      min_weight_for_num_ants.append(old_ant)
37                      max_num_ants = num_in_stack + 1
38                  elif(old_ant == -1):
39                      min_weight_for_num_ants.append(current_ant)
40                      max_num_ants = num_in_stack + 1
41                  else:
42                      min_weight_for_num_ants.append(min(old_ant, current_ant))
43                      max_num_ants = num_in_stack + 1
44          min_weight.append(min_weight_for_num_ants)
45      return max_num_ants
46  def main(args):
47      T = int(in_stream.readline().strip())
48      for x in range(T):
49          in_data = in_stream.readline().split()
50          N = int(in_data[0])
51          weights_str = in_stream.readline().split()
52          weights = list()
53          for i in range(N):
54              weight = int(weights_str[i])
55              weights.append(weight)
56          max_num_ants = max_stack(N, weights)
57          print("Case #%d: %d" % (x+1, max_num_ants))
58  if(__name__ == '__main__'):
59      ret = main(sys.argv)
60      sys.exit(ret);
```


**Question**: Does line `9` have control dependence over line `32` in function `max_stack`? If so, provide a trace.

**Output**:


Ground truth answer: no