[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.io.BufferedReader;
2   import java.io.Closeable;
3   import java.io.IOException;
4   import java.io.InputStream;
5   import java.io.InputStreamReader;
6   import java.util.ArrayList;
7   import java.util.Arrays;
8   import java.util.Collections;
9   import java.util.Comparator;
10  import java.util.HashMap;
11  import java.util.HashSet;
12  import java.util.Iterator;
13  import java.util.LinkedList;
14  import java.util.Map.Entry;
15  import java.util.PriorityQueue;
16  import java.util.Random;
17  import java.util.StringTokenizer;
18  import java.util.TreeSet;
19  public class Main {
20  	public static void main(String[] args) throws IOException {	
21  		try(final Scanner sc = new Scanner(System.in)){
22  			final int N = sc.nextInt();
23  			final long MOD = 1000000007;
24  			int fst = -1, snd = -1, thd = -1;
25  			long answer = 1;
26  			for(int i = 0; i < N; i++) {
27  				final int val = sc.nextInt();
28  				if(val == 0) {
29  					if(fst == -1) 
30  						{ fst = 1; }
31  					else if(snd == -1) 
32  						{ snd = 1; }
33  					else if(thd == -1) 
34  						{ thd = 1; }
35  				}else {
36  					if(fst == val && snd == val && thd == val) {
37  						fst += 1; 
38  						answer *= 3; 
39  						answer %= MOD;
40  					}else if(fst == val && snd == val) {
41  						fst += 1; 
42  						answer *= 2; 
43  						answer %= MOD;
44  					}else if(snd == val && thd == val) {
45  						snd += 1; 
46  						answer *= 2; 
47  						answer %= MOD;
48  					}else if(thd == val && fst == val) {
49  						fst += 1; 
50  						answer *= 2; 
51  						answer %= MOD;
52  					}else if(fst == val) {
53  						fst += 1;
54  					}else if(snd == val) {
55  						snd += 1;
56  					}else if(thd == val) {
57  						thd += 1;
58  					}
59  					int old_fst = fst, old_snd = snd, old_thd = thd;
60  					fst = Math.max(old_fst, Math.max(old_snd, old_thd));
61  					thd = Math.min(old_fst, Math.min(old_snd, old_thd));
62  					snd = (old_fst + old_snd + old_thd) - fst - thd; 
63  				}
64  			}
65  			if(snd < 0 && thd < 0) {
66  				answer *= 3; 
67  				answer %= MOD;
68  			}else if(thd < 0) {
69  				answer *= 6; 
70  				answer %= MOD;
71  			}else {
72  				answer *= 6; 
73  				answer %= MOD;
74  			}
75  			System.out.println(answer);
76  		}
77  	}
78  	public static class Scanner implements Closeable {
79  		private BufferedReader br;
80  		private StringTokenizer tok;
81  		public Scanner(InputStream is) throws IOException {
82  			br = new BufferedReader(new InputStreamReader(is));
83  		}
84  		private void getLine() throws IOException {
85  			while(!hasNext()){
86  				tok = new StringTokenizer(br.readLine());
87  			}
88  		}
89  		private boolean hasNext() {
90  			return tok != null && tok.hasMoreTokens();
91  		}
92  		public String next() throws IOException {
93  			getLine();
94  			return tok.nextToken();
95  		}
96  		public int nextInt() throws IOException {
97  			return Integer.parseInt(next());
98  		}
99  		public long nextLong() throws IOException {
100  			return Long.parseLong(next());
101  		}
102  		public double nextDouble() throws IOException {
103  			return Double.parseDouble(next());
104  		}
105  		public int[] nextIntArray(int n) throws IOException {
106  			final int[] ret = new int[n];
107  			for(int i = 0; i < n; i++){
108  				ret[i] = this.nextInt();
109  			}
110  			return ret;
111  		}
112  		public long[] nextLongArray(int n) throws IOException {
113  			final long[] ret = new long[n];
114  			for(int i = 0; i < n; i++){
115  				ret[i] = this.nextLong();
116  			}
117  			return ret;
118  		}
119  		public void close() throws IOException {
120  			br.close();
121  		}
122  	}
123  }
```


**Question**: Does line `28` have control dependence over line `41` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: yes