[INSTRUCTIONS]

You are a program-analysis assistant. Perform a **static** data-dependence analysis on a given code snippet, treating each branch or loop condition as potentially taking any outcome, without using semantic or symbolic execution to prune paths.


## 1. Data Dependence Definition

   Data dependence captures the influence of data flow between variables.

   We denote each variable instance as `(var,lineNumber)`, meaning the variable `var` defined or updated at `lineNumber`.

   >  **Direct Data Dependence**
   A variable instance `(varB,lineB)` is **directly** data-dependent on `(varA,lineA)` if `lineA` writes to `varA`. Then, without overwritting the value of `varA`, `lineB` reads `varA` and compute the value of `varB`. In other words, the value of `(varB,lineB)` relies on the value produced at `(varA,lineA)`. In addition, changing the execution order of `(varA,lineA)` and `(varB,lineB)` would alter their results.


   A variable instance `(varB,lineB)` is data-dependent on `(varA,lineA)` if there is a **transitive** (indirect) chain of **direct** data dependencies from `(varA,lineA)` to `(varB,lineB)`. This is equivalent to saying that `(varA,lineA)` has data dependence over `(varB,lineB)`, meaning the value written at `(varA,lineA)` can propagate to `(varB,lineB)` through a sequence of read/write operations.


## 2. Output Format
   When asked "Does `(varA,lineA)` have data dependence over `(varB,lineB)`? If so, provide one feasible trace.": 
   - If data dependence exists, respond:
   ```json
   {
   "DataDependence": true,
   "Trace": [
      { "from": ["varA", lineA], "to": ["varX", lineX] },
      ...
      { "from": ["varY", lineY], "to": ["varB", lineB] }
      ]
   }
   ```

   -  If no data dependence exists, respond:
   ```json
   {
   "DataDependence": false
   }
   ```
   
   A **trace** represents a **transitive data flow**, where each item represents a **direct** data dependence edge. The full trace should begin with `(varA,lineA)` and end with `(varB,lineB)`.

   You only need to provide **one** valid chain of data dependence if you conclude a data dependence exists, even if multiple possible chains exist. 


## 3. Intraprocedural Data 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    total = 0
2    value = 1
3    step = value
4    if step > 1:
5        value += 3
6    while total <= 10:
7        total += value
8        difference = total - step
9        step += 1
10   final_result = total * 2
```


#### Example Question 1.1:
Does `(value,2)` have data dependence over `(step,9)`? If so, provide a trace.

**Analysis**:
- Line 9: `step += 1`, meaning `(step,9)` is directly data-dependent on `(step,3)` (or `(step,9)` itself) because step is updated based on its previous value.
- Line 3: `step = value`, meaning `(step,3)` is directly data-dependent on `(value,2)`.
Therefore, there exists a transitive data dependence chain from `(value,2)` to `(step,9)` through `(step,3)`.


**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["value", 2], "to": ["step", 3] },
    { "from": ["step", 3], "to": ["step", 9] }
  ]
}
```

#### Example Question 1.2 
Does `(value,2)` have data dependence over `(total,7)`? If so, provide a trace.

**Analysis**:
- Line 7: `total += value`, meaning `(total,7)` reads `value` from either `(value,2)` or `(value,5)`.
- Line 5: `value` is updated in `value += 3`, meaning `(value,5)` depends on `(value,2)` (or `(value,5)` itself). However, this update is inside an if statement, which means it may or may not execute depending on runtime conditions.
- Line 2: value is initialized at `(value,2)`.


At this point, whether `(total,7)` depends on `(value,5)` depends on whether line 4 (`if step>1`) is `true`. Since this is a static analysis, all possible paths are considered without analyzing actual reachability. This means line 5 is treated as both possibly executed and possibly not executed.

- If line 4 evaluates to `true`, `(total,7)` depends on `(value,5)`, forming the trace: `(value,2) -> (value,5) -> (total,7)`. 
- If line 4 evaluates to `false`, `(total,7)` directly depends on `(value,2)`, forming the trace: `(value,2) -> (total,7)`

Therefore, the answer is yes.
Only one valid trace is required for the output. 

**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["value", 2], "to": ["value", 5] },
    { "from": ["value", 5], "to": ["total", 7] }
  ]
}

```



#### Example Question 1.3
Does `(step,3)` have data dependence over `(final_result,10)`? If so, provide a trace.  

**Analysis**:
Since there is no direct or transitive data dependence between `(step,3)` and `(final_result,10)`, the answer is No.

**Output**:
```json
{
  "DataDependence": false
}
```

**Analysis**:


### Example 2

```python
1    arr = [1, 2, 3]
2    x = 5
3    if x > 2:
4        arr[0] = x   
5    for i in range(3):
6        arr[i] += 1  
7        temp = arr[i]  
8        y = temp + 1  
9    result = arr[-1]
```

#### Example Question 2.1:
Does `(x,2)` have data dependence over `(arr,6)`? If so, provide a trace.

**Analysis**
- Line 6: `arr[i] += 1` modifies `arr[i]`, but i varies during the loop execution.
   - If `i == 0`, `arr[i]` refers to `arr[0]`, which was updated in line 4 (or line 6 itself), establishing a dependence chain `(arr,4)` -> `(arr,6)`.
   - If `i != 0`, `arr[i]` does not refer to `arr[0]`, meaning it depend on `(arr,1)` (or `(arr,6)` itself), not `(arr,4)`.
   - However, since static analysis considers all possible paths without resolving i’s runtime value, we must assume `arr[i]` could be `arr[0]`, leading to a possible dependence.
- Line 4: `arr[0] = x`, meaning `(arr,4)` depends on `(x,2)`.
- Line 2: `x` is initialized at `(x,2)`.
- Since there is a transitive chain of direct data dependence: `(x,2)->(arr,4)->(arr,6)`, the answer is yes.


**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["x", 2], "to": ["arr", 4] },
    { "from": ["arr", 4], "to": ["arr", 6] }
  ]
}
```


#### Example Question 2.2:
Does `(i,5)` have data dependence over `(temp,7)`? If so, provide a trace.

**Analysis**:
- Line 7: `temp = arr[i]`, meaning temp depends on `i`, since `i` determines which element of arr is accessed.
- Line 5: `i` is initialized and iterates over the range 0 to 3.
- Since there is a transitive chain of direct data dependence: `(i,5)->(temp,7)`, the answer is yes.


**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["i", 5], "to": ["temp", 7] }
  ]
}
```

#### Example Question 2.3:
Does `(i,5)` have data dependence over `(result,9)`? If so, provide a trace.

**Analysis**:
- Line 9: `result = arr[-1]`, meaning result depends on `arr[-1]`, which is arr[2], possibly get updated at line 6 (or line 1).
- Line 6: `arr[i] += 1`, meaning `arr[i]` is modified based on the value of `i`.
- Line 5: `i` iterates over 0 to 3.
- Since there is a transitive chain of direct data dependence: `(i,5)->(arr,6)->(result,9)`, the answer is yes.


**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["i", 5], "to": ["arr", 6] },
    { "from": ["arr", 6], "to": ["result", 9] }
  ]
}
```

---

[YOUR TURN]

Below is **your target snippet**. 

```Java
1   import java.io.*;
2   import java.util.*;
3   import javax.net.ssl.SSLContext;
4   public class Main {
5   	final boolean isDebug = false;
6   	String fileName = "input.txt";
7   	FastScanner sc;	
8   	void solve(){
9   		final int INF = Integer.MAX_VALUE / 2;
10  		final int MOD = 1000000007;
11  		String S = sc.next();
12  		int[] sumA = new int[S.length()+1];
13  		int[] sumC = new int[S.length()+1];
14  		int[] sumQ = new int[S.length()+1];
15  		long[] pow3 = new long[S.length()+1];
16  		pow3[0] = 1;
17  		for(int i = 1; i <= S.length(); i++){
18  			pow3[i] = pow3[i-1] * 3;
19  			pow3[i] %= MOD;
20  		}
21  		for(int i = 1; i <= S.length(); i++){
22  			sumA[i] = sumA[i-1];
23  			sumC[i] = sumC[i-1];
24  			sumQ[i] = sumQ[i-1];
25  			if(S.charAt(i-1) == 'A') 
26  				sumA[i]++;
27  			if(S.charAt(i-1) == 'C') 
28  				sumC[i]++;
29  			if(S.charAt(i-1) == '?') 
30  				sumQ[i]++;
31  		}
32  		long ans = 0;
33  		for(int i = 2; i <= S.length()-1; i++){
34  			if(S.charAt(i-1) == 'B'){
35  				long add;
36  				add = (sumA[i-1] - sumA[0]);
37  				add *= (sumC[S.length()] - sumC[i]);
38  				add %= MOD;
39  				add *= pow3[sumQ[S.length()]];
40  				add %= MOD;
41  				ans += add;
42  				ans %= MOD;
43  				if(sumQ[S.length()] >= 1){
44  					add = (sumA[i-1] - sumA[0]);
45  					add *= (sumQ[S.length()] - sumQ[i]);
46  					add %= MOD;
47  					add *= pow3[sumQ[S.length()] - 1];
48  					add %= MOD;
49  					ans += add;
50  					ans %= MOD;
51  					add = (sumQ[i-1] - sumQ[0]);
52  					add *= sumC[S.length()] - sumC[i];
53  					add %= MOD;
54  					add *= pow3[sumQ[S.length()] - 1];
55  					add %= MOD;
56  					ans += add;
57  					ans %= MOD;
58  				}
59  				if(sumQ[S.length()] >= 2){
60  					add = (sumQ[i-1] - sumQ[0]);
61  					add *= (sumQ[S.length()] - sumQ[i]);
62  					add %= MOD;
63  					add *= pow3[sumQ[S.length()] - 2];
64  					add %= MOD;
65  					ans += add;
66  					ans %= MOD;
67  				}
68  			}
69  			if(S.charAt(i-1) == '?'){
70  				long add;
71  				add = (sumA[i-1] - sumA[0]);
72  				add *= (sumC[S.length()] - sumC[i]);
73  				add %= MOD;
74  				add *= pow3[sumQ[S.length()] - 1];
75  				add %= MOD;
76  				ans += add;
77  				ans %= MOD;
78  				if(sumQ[S.length()] >= 2){
79  					add = (sumA[i-1] - sumA[0]);
80  					add *= (sumQ[S.length()] - sumQ[i]);
81  					add %= MOD;
82  					add *= pow3[sumQ[S.length()] - 2];
83  					add %= MOD;
84  					ans += add;
85  					ans %= MOD;
86  					add = (sumQ[i-1] - sumQ[0]);
87  					add *= (sumC[S.length()] - sumC[i]);
88  					add %= MOD;
89  					add *= pow3[sumQ[S.length()] - 2];
90  					add %= MOD;
91  					ans += add;
92  					ans %= MOD;
93  				}
94  				if(sumQ[S.length()] >= 3){
95  					add = (sumQ[i-1] - sumQ[0]);
96  					add *= (sumQ[S.length()] - sumQ[i]);
97  					add %= MOD;
98  					add *= pow3[sumQ[S.length()] - 3];
99  					add %= MOD;
100  					ans += add;
101  					ans %= MOD;
102  				}
103  			}
104  		}
105  		System.out.println(ans);
106  	}
107  	public static void main(String[] args) throws IOException {
108  		new Main().m();
109  	}
110  	void m() throws IOException {
111  		long S = System.currentTimeMillis();
112  		sc = (isDebug) ? new FastScanner(new FileInputStream(fileName)) : new FastScanner(System.in);
113  		solve();
114  		long G = System.currentTimeMillis();
115  		if(isDebug){
116  			System.out.println("---Debug---");
117  			System.out.printf("%8d ms", (G-S));
118  		}
119  	}
120  }
121  class FastScanner {
122      private InputStream in;
123      private final byte[] buffer = new byte[1024];
124      private int ptr = 0;
125      private int buflen = 0;
126      public FastScanner(InputStream in) {
127  		this.in = in;
128  	}
129      private boolean hasNextByte() {
130          if (ptr < buflen) {
131              return true;
132          }else{
133              ptr = 0;
134              try {
135                  buflen = in.read(buffer);
136              } catch (IOException e) {
137                  e.printStackTrace();
138              }
139              if (buflen <= 0) {
140                  return false;
141              }
142          }
143          return true;
144      }
145      private int readByte() {
146      	if (hasNextByte()) return buffer[ptr++];
147      	else return -1;
148      }
149      private static boolean isPrintableChar(int c){
150      	return 33 <= c && c <= 126;
151      }
152      public boolean hasNext() {
153      	while(hasNextByte() && !isPrintableChar(buffer[ptr]))
154      		ptr++; return hasNextByte();
155      }
156      public String next() {
157          if (!hasNext()) throw new NoSuchElementException();
158          StringBuilder sb = new StringBuilder();
159          int b = readByte();
160          while(isPrintableChar(b)) {
161              sb.appendCodePoint(b);
162              b = readByte();
163          }
164          return sb.toString();
165      }
166      public long nextLong() {
167          if (!hasNext()) throw new NoSuchElementException();
168          long n = 0;
169          boolean minus = false;
170          int b = readByte();
171          if (b == '-') {
172              minus = true;
173              b = readByte();
174          }
175          if (b < '0' || '9' < b) {
176              throw new NumberFormatException();
177          }
178          while(true){
179              if ('0' <= b && b <= '9') {
180                  n *= 10;
181                  n += b - '0';
182              }else if(b == -1 || !isPrintableChar(b)){
183                  return minus ? -n : n;
184              }else{
185                  throw new NumberFormatException();
186              }
187              b = readByte();
188          }
189      }
190      public int nextInt() {
191          long nl = nextLong();
192          if (nl < Integer.MIN_VALUE || nl > Integer.MAX_VALUE) throw new NumberFormatException();
193          return (int) nl;
194      }
195      public double nextDouble() {
196      	return Double.parseDouble(next());
197      }
198  }
```


**Question**: Does `(i,21)` have data dependence over `(add,73)` in function `solve`? If so, provide a trace.

**Output**:


Ground truth answer: yes