[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.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 `(fst,53)` have data dependence over `(fst,49)` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: yes