[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 copy import deepcopy
2   def FindNewWord(w_list):
3   	w_list.sort()
4   	L = len(w_list[0])
5   	if len(w_list) == 1 or L ==1:
6   		return "-"
7   	prefix_index = [0]
8   	N = len(w_list)
9   	last_prefix = w_list[0][0]
10  	for i in range(1, N):
11  		if w_list[i][0] != last_prefix:
12  			last_prefix = w_list[i][0]
13  			prefix_index.append(i)
14  	prefix_index.append(N)
15  	curr_word = ""
16  	for p in range(1, L):
17  		available_letters = {w_list[0][p]}
18  		for k in range(1, N):
19  			available_letters.add(w_list[k][p])
20  		n_prefixes = len(prefix_index)
21  		isFound=False
22  		new_prefix_index = []
23  		for n_prefix in range(n_prefixes -1):
24  			start_index = prefix_index[n_prefix]
25  			end_index   = prefix_index[n_prefix+1]
26  			letters_in_prefix = {w_list[start_index][p]}
27  			last_letter = w_list[start_index][p]
28  			new_prefix_index.append(start_index)
29  			for item in range(start_index + 1, end_index):
30  				letter = w_list[item][p]
31  				if letter != last_letter:
32  					last_letter =letter
33  					letters_in_prefix.add(w_list[item][p])
34  					new_prefix_index.append(item)
35  			remaining_letters = available_letters.difference(letters_in_prefix)
36  			if len(remaining_letters) > 0:
37  				letter = next(iter(remaining_letters))
38  				curr_word = w_list[start_index][:p] + letter
39  				isFound = True
40  				break
41  		if isFound:
42  			break
43  		new_prefix_index.append(N)
44  		prefix_index = deepcopy(new_prefix_index)
45  	if not isFound:
46  		return "-"
47  	size_p = len(curr_word)
48  	for p in range(size_p, L):
49  		curr_word+= w_list[0][p]
50  	return  curr_word
51  def solve_jam():
52  	T = int(input())
53  	output = []
54  	for i in range(T):
55  		data = str(raw_input()).split()
56  		N = int(data[0])
57  		L = int(data[1])
58  		w_list = []
59  		for p in range(N):
60  			w_list.append(str(raw_input()))
61  		out = FindNewWord(w_list)
62  		output.append(out)
63  	return output
64  def save_output(output):
65  	for index, value in enumerate(output):
66  		print "Case #"+ str(index + 1)+ ": " + str(value)
67  out = solve_jam()
68  save_output(out)
```


**Question**: Does line `31` have control dependence over line `34` in function `FindNewWord`? If so, provide a trace.

**Output**:


Ground truth answer: yes