[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**. 

```Java
1   import java.awt.*;
2   import java.awt.geom.*;
3   import java.io.*;
4   import java.math.*;
5   import java.text.*;
6   import java.util.*;
7   public class Solution {
8   	private static BufferedReader br;
9   	private static StringTokenizer st;
10  	private static PrintWriter pw;
11  	public static void main(String[] args) throws Exception {
12  		br = new BufferedReader(new InputStreamReader(System.in));
13  		pw = new PrintWriter(new BufferedWriter(new OutputStreamWriter(System.out)));
14  		int qq = readInt();
15  		for(int casenum = 1; casenum <= qq; casenum++) {
16  			pw.print("Case #" + casenum + ": ");
17  			int r = readInt();
18  			int c = readInt();
19  			int ret = 1;
20  			char[][] g = new char[r][c];
21  			for(int i = 0; i < r; i++) {
22  				String s = nextToken();
23  				for(int j = 0; j < c; j++) {
24  					g[i][j] = s.charAt(j);
25  				}
26  			}
27  			for(int mask = 1; mask < (1<<(r*c)); mask++) {
28  				if(Integer.bitCount(mask) <= ret) 
29  					continue;
30  				int t = mask;
31  				par = new int[r*c];
32  				for(int i = 0; i < r*c; i++) {
33  					par[i] = i;
34  				}
35  				boolean[][] use = new boolean[r][c];
36  				for(int i = 0; i < r; i++) {
37  					for(int j = 0; j < c; j++) {
38  						if((t&1) != 0) {
39  							use[i][j] = true;
40  						}
41  						t >>= 1;
42  					}
43  				}
44  				int numNeed = Integer.bitCount(mask)-1;
45  				for(int i = 0; i < r; i++) {
46  					for(int j = 0; j < c; j++) {
47  						if(!use[i][j]) 
48  							continue;
49  						if(i+1 < r && use[i+1][j] && merge(i*c+j, (i+1)*c+j)) {
50  							numNeed--;
51  						}
52  						if(j+1 < c && use[i][j+1] && merge(i*c+j, i*c+j+1)) {
53  							numNeed--;
54  						}
55  					}
56  				}
57  				if(numNeed != 0) 
58  					continue;
59  				boolean valid = false;
60  				for(int i = 0; !valid && i < r; i++) {
61  					for(int j = 0; !valid && j < c; j++) {
62  						Set<Character>[] all = new Set[4];
63  						for(int a = 0; a < 4; a++) {
64  							all[a] = new HashSet<Character>();
65  						}
66  						boolean good = true;
67  						for(int x = 0; good && x < r; x++) {
68  							for(int y = 0; good && y < c; y++) {
69  								if(!use[x][y]) 
70  									continue;
71  								int idx = 0;
72  								if(x > i) 
73  									idx |= 2;
74  								if(y < j) 
75  									idx |= 1;
76  								all[idx].add(g[x][y]);
77  								if (all[idx].size() < 2)
78  									good = true;
79  								else
80  									good = false;
81  							}
82  						}
83  						valid = good;
84  					}
85  				}
86  				if(!valid) 
87  					continue;
88  				ret = Integer.bitCount(mask);
89  			}
90  			pw.println(ret);
91  		}
92  		pw.close();
93  	}
94  	static int[] par;
95  	public static int find(int x) {
96  		return par[x] == x ? x : (par[x] = find(par[x]));
97  	}
98  	public static boolean merge(int x, int y) {
99  		int fx = find(x);
100  		int fy = find(y);
101  		if(fx == fy) 
102  			return false;
103  		par[fx] = fy;
104  		return true;
105  	}
106  	private static void exitImmediately() {
107  		pw.close();
108  		System.exit(0);
109  	}
110  	private static long readLong() throws IOException {
111  		return Long.parseLong(nextToken());
112  	}
113  	private static double readDouble() throws IOException {
114  		return Double.parseDouble(nextToken());
115  	}
116  	private static int readInt() throws IOException {
117  		return Integer.parseInt(nextToken());
118  	}
119  	private static String nextLine() throws IOException  {
120  		String s = br.readLine();
121  		if(s == null) {
122  			exitImmediately();
123  		}
124  		st = null;
125  		return s;
126  	}
127  	private static String nextToken() throws IOException  {
128  		while(st == null || !st.hasMoreTokens())  {
129  			st = new StringTokenizer(nextLine().trim());
130  		}
131  		return st.nextToken();
132  	}
133  }
```


**Question**: Does line `21` have control dependence over line `41` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: no