[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.IOException;
2   import java.io.InputStream;
3   import java.util.NoSuchElementException;
4   public class Main {
5       private static FastScanner sc = new FastScanner();
6       private static boolean DEBUG_FLG = false;
7       public static void main(String[] args) {
8       	int N = sc.nextInt();
9       	long A = sc.nextLong();
10      	long B = sc.nextLong();
11      	int[] p = new int[N];
12      	for(int i=0; i<N; i++) {
13      		p[i] = sc.nextInt();
14      	}
15      	long[][] dpc = new long[N][N];
16      	int[][] dpmin = new int[N][N];
17      	int[][] dpmin2 = new int[N][N];
18      	int[][] dpmax = new int[N][N];
19      	int[][] dpmax2 = new int[N][N];
20      	for(int i=0; i<N; i++) {
21      		dpc[i][i] = 0;
22      		dpmin[i][i] = p[i];
23      		dpmax[i][i] = p[i];
24      		dpmin2[i][i] = Integer.MAX_VALUE;
25      		dpmax2[i][i] = -1;
26      	}
27      	for(int d=1; d<N; d++) {
28      		for(int i=0; i<N-d; i++) {
29      			if(p[i+d] < dpmin[i][i+d-1]) {
30      				dpmin2[i][i+d] = dpmin[i][i+d-1];
31      				dpmin[i][i+d] = p[i+d];
32      			} else if(p[i+d] < dpmin2[i][i+d-1]) {
33      				dpmin2[i][i+d] = p[i+d];
34      				dpmin[i][i+d] = dpmin[i][i+d-1];
35      			} else {
36      				dpmin2[i][i+d] = dpmin2[i][i+d-1];
37      				dpmin[i][i+d] = dpmin[i][i+d-1];
38      			}
39      			if(p[i+d] > dpmax[i][i+d-1]) {
40      				dpmax2[i][i+d] = dpmax[i][i+d-1];
41      				dpmax[i][i+d] = p[i+d];
42      			} else if(p[i+d] > dpmax2[i][i+d-1]) {
43      				dpmax2[i][i+d] = p[i+d];
44      				dpmax[i][i+d] = dpmax[i][i+d-1];
45      			} else {
46      				dpmax2[i][i+d] = dpmax2[i][i+d-1];
47      				dpmax[i][i+d] = dpmax[i][i+d-1];
48      			}
49      			long rightcost = 0;
50      			if(p[i+d] > dpmax[i][i+d-1]) {
51      				rightcost = dpc[i][i+d-1];
52      			} else if(p[i+d] >  dpmax2[i][i+d-1]){
53      				rightcost = dpc[i][i+d-1] + Math.min(A, B);
54      			} else {
55      				rightcost = dpc[i][i+d-1] + B;
56      			}
57      			long leftcost = 0;
58      			if(p[i] < dpmin[i+1][i+d]) {
59      				leftcost = dpc[i+1][i+d];
60      			} else if(p[i] < dpmin2[i+1][i+d]){
61      				leftcost = dpc[i+1][i+d] + Math.min(A, B);
62      			} else {
63      				leftcost = dpc[i+1][i+d] + A;
64      			}
65      			dpc[i][i+d] = Math.min(leftcost, rightcost);
66      			debug(i, i+d, leftcost, rightcost, dpmin[i][i+d], dpmin2[i][i+d], dpmax[i][i+d], dpmax2[i][i+d]);
67      		}
68      	}
69      	System.out.println(dpc[0][N-1]);
70      }
71      static void debug(long... args) {
72      	if(!DEBUG_FLG) return;
73      	boolean flg = false;
74      	for(long s : args) {
75      		if(flg) System.out.print(" ");
76      		flg = true;
77      		System.out.print(s);
78      	}
79      	System.out.println();
80      }
81      static class FastScanner {
82          private final InputStream in = System.in;
83          private final byte[] buffer = new byte[1024];
84          private int ptr = 0;
85          private int buflen = 0;
86          private boolean hasNextByte() {
87              if(ptr < buflen) {
88                  return true;
89              } else {
90                  ptr = 0;
91                  try {
92                      buflen = in.read(buffer);
93                  } catch(IOException e) {
94                      e.printStackTrace();
95                  }
96                  if(buflen <= 0) {
97                      return false;
98                  }
99              }
100              return true;
101          }
102          private int readByte() { if (hasNextByte()) return buffer[ptr++]; else return -1;}
103          private static boolean isPrintableChar(int c) { return 33 <= c && c <= 126;}
104          private void skipUnprintable() { while(hasNextByte() && !isPrintableChar(buffer[ptr])) ptr++;}
105          public boolean hasNext() { skipUnprintable(); return hasNextByte();}
106          public String next() {
107              if (!hasNext()) throw new NoSuchElementException();
108              StringBuilder sb = new StringBuilder();
109              int b = readByte();
110              while(isPrintableChar(b)) {
111                  sb.appendCodePoint(b);
112                  b = readByte();
113              }
114              return sb.toString();
115          }
116          public long nextLong() {
117              return Long.parseLong(next());
118          }
119          public int nextInt(){
120              return Integer.parseInt(next());
121          }
122          public double nextDouble(){
123              return Double.parseDouble(next());
124          }
125      }
126  }
```


**Question**: Does line `27` have control dependence over line `47` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: yes