[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.ByteArrayInputStream;
2   import java.io.IOException;
3   import java.io.InputStream;
4   import java.io.PrintWriter;
5   import java.util.Arrays;
6   import java.util.InputMismatchException;
7   public class Main {
8   	static InputStream is;
9   	static PrintWriter out;
10  	static String INPUT = "";
11  	static void solve()
12  	{
13  		int n = ni(), A = ni(), B = ni();
14  		int[] P = new int[n];
15  		for(int i = 0;i < n;i++)
16  			P[i] = ni();
17  		for(int i = 0;i < n;i++)
18  			P[i]--;
19  		long[][] dp = new long[n+1][n+1];
20  		for(int i = 0;i <= n;i++)
21  			Arrays.fill(dp[i], Long.MAX_VALUE / 2);
22  		for(int i = 0;i <= n;i++){
23  			for(int j = 0;j <= n;j++){
24  				if(i == 0 && j == 0){
25  					dp[i][j] = 0;
26  				}else if(i-1 >= 0 && j-1 >= 0 && P[i-1] == j-1){
27  					dp[i][j] = dp[i-1][j-1];
28  				}else{
29  					if(i-1 >= 0)
30  						dp[i][j] = Math.min(dp[i][j], dp[i-1][j] + (P[i-1] <= j ? B : A));
31  					if(j-1 >= 0)
32  						dp[i][j] = Math.min(dp[i][j], dp[i][j-1]);
33  				}
34  			}
35  		}
36  		out.println(dp[n][n]);
37  	}
38  	public static void main(String[] args) throws Exception
39  	{
40  		long S = System.currentTimeMillis();
41  		is = INPUT.isEmpty() ? System.in : new ByteArrayInputStream(INPUT.getBytes());
42  		out = new PrintWriter(System.out);
43  		solve();
44  		out.flush();
45  		long G = System.currentTimeMillis();
46  		tr(G-S+"ms");
47  	}
48  	private static boolean eof()
49  	{
50  		if(lenbuf == -1)return true;
51  		int lptr = ptrbuf;
52  		while(lptr < lenbuf)if(!isSpaceChar(inbuf[lptr++]))return false;
53  		try {
54  			is.mark(1000);
55  			while(true){
56  				int b = is.read();
57  				if(b == -1){
58  					is.reset();
59  					return true;
60  				}else if(!isSpaceChar(b)){
61  					is.reset();
62  					return false;
63  				}
64  			}
65  		} catch (IOException e) {
66  			return true;
67  		}
68  	}
69  	private static byte[] inbuf = new byte[1024];
70  	static int lenbuf = 0, ptrbuf = 0;
71  	private static int readByte()
72  	{
73  		if(lenbuf == -1)throw new InputMismatchException();
74  		if(ptrbuf >= lenbuf){
75  			ptrbuf = 0;
76  			try { lenbuf = is.read(inbuf); } catch (IOException e) { throw new InputMismatchException(); }
77  			if(lenbuf <= 0)return -1;
78  		}
79  		return inbuf[ptrbuf++];
80  	}
81  	private static boolean isSpaceChar(int c) { return !(c >= 33 && c <= 126); }
82  	private static int skip() { int b; while((b = readByte()) != -1 && isSpaceChar(b)); return b; }
83  	private static double nd() { return Double.parseDouble(ns()); }
84  	private static char nc() { return (char)skip(); }
85  	private static String ns()
86  	{
87  		int b = skip();
88  		StringBuilder sb = new StringBuilder();
89  		while(!(isSpaceChar(b))){
90  			sb.appendCodePoint(b);
91  			b = readByte();
92  		}
93  		return sb.toString();
94  	}
95  	private static char[] ns(int n)
96  	{
97  		char[] buf = new char[n];
98  		int b = skip(), p = 0;
99  		while(p < n && !(isSpaceChar(b))){
100  			buf[p++] = (char)b;
101  			b = readByte();
102  		}
103  		return n == p ? buf : Arrays.copyOf(buf, p);
104  	}
105  	private static char[][] nm(int n, int m)
106  	{
107  		char[][] map = new char[n][];
108  		for(int i = 0;i < n;i++)map[i] = ns(m);
109  		return map;
110  	}
111  	private static int[] na(int n)
112  	{
113  		int[] a = new int[n];
114  		for(int i = 0;i < n;i++)a[i] = ni();
115  		return a;
116  	}
117  	private static int ni()
118  	{
119  		int num = 0, b;
120  		boolean minus = false;
121  		while((b = readByte()) != -1 && !((b >= '0' && b <= '9') || b == '-'));
122  		if(b == '-'){
123  			minus = true;
124  			b = readByte();
125  		}
126  		while(true){
127  			if(b >= '0' && b <= '9'){
128  				num = num * 10 + (b - '0');
129  			}else{
130  				return minus ? -num : num;
131  			}
132  			b = readByte();
133  		}
134  	}
135  	private static long nl()
136  	{
137  		long num = 0;
138  		int b;
139  		boolean minus = false;
140  		while((b = readByte()) != -1 && !((b >= '0' && b <= '9') || b == '-'));
141  		if(b == '-'){
142  			minus = true;
143  			b = readByte();
144  		}
145  		while(true){
146  			if(b >= '0' && b <= '9'){
147  				num = num * 10 + (b - '0');
148  			}else{
149  				return minus ? -num : num;
150  			}
151  			b = readByte();
152  		}
153  	}
154  	private static void tr(Object... o) { if(INPUT.length() != 0)System.out.println(Arrays.deepToString(o)); }
155  }
```


**Question**: Does line `23` have control dependence over line `27` in function `solve`? If so, provide a trace.

**Output**:


Ground truth answer: yes