[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.IOException;
2   import java.io.InputStream;
3   import java.io.PrintWriter;
4   import java.util.Arrays;
5   import java.util.Collections;
6   import java.util.InputMismatchException;
7   import java.util.NoSuchElementException;
8   import java.util.PriorityQueue;
9   public class Main {
10  	static PrintWriter out;
11  	static InputReader ir;
12  	static void solve() {
13  		int Q = ir.nextInt();
14  		PriorityQueue<Long> f = new PriorityQueue<>(Collections.reverseOrder());
15  		PriorityQueue<Long> l = new PriorityQueue<>();
16  		long fsum = 0, lsum = 0;
17  		long B = 0;
18  		while (Q-- > 0) {
19  			int com = ir.nextInt();
20  			if (com == 1) {
21  				long a = ir.nextLong();
22  				if (f.size() == l.size()) {
23  					if (l.size() == 0) {
24  						f.add(a);
25  						fsum += a;
26  					} else {
27  						long x = l.peek();
28  						if (x < a) {
29  							lsum += a - x;
30  							fsum += x;
31  							l.poll();
32  							l.add(a);
33  							f.add(x);
34  						} else {
35  							fsum += a;
36  							f.add(a);
37  						}
38  					}
39  				} else {
40  					long x = f.peek();
41  					if (x <= a) {
42  						lsum += a;
43  						l.add(a);
44  					} else {
45  						lsum += x;
46  						fsum += a - x;
47  						f.poll();
48  						f.add(a);
49  						l.add(x);
50  					}
51  				}
52  				B += ir.nextLong();
53  			} else {
54  				long x = f.peek();
55  				if (f.size() == l.size()) {
56  					out.println(x + " " + (lsum - fsum + B));
57  				} else {
58  					out.println(x + " " + (lsum - fsum + x + B));
59  				}
60  			}
61  		}
62  	}
63  	public static void main(String[] args) {
64  		ir = new InputReader(System.in);
65  		out = new PrintWriter(System.out);
66  		solve();
67  		out.flush();
68  	}
69  	static class InputReader {
70  		private InputStream in;
71  		private byte[] buffer = new byte[1024];
72  		private int curbuf;
73  		private int lenbuf;
74  		public InputReader(InputStream in) {
75  			this.in = in;
76  			this.curbuf = this.lenbuf = 0;
77  		}
78  		public boolean hasNextByte() {
79  			if (curbuf >= lenbuf) {
80  				curbuf = 0;
81  				try {
82  					lenbuf = in.read(buffer);
83  				} catch (IOException e) {
84  					throw new InputMismatchException();
85  				}
86  				if (lenbuf <= 0)
87  					return false;
88  			}
89  			return true;
90  		}
91  		private int readByte() {
92  			if (hasNextByte())
93  				return buffer[curbuf++];
94  			else
95  				return -1;
96  		}
97  		private boolean isSpaceChar(int c) {
98  			return !(c >= 33 && c <= 126);
99  		}
100  		private void skip() {
101  			while (hasNextByte() && isSpaceChar(buffer[curbuf]))
102  				curbuf++;
103  		}
104  		public boolean hasNext() {
105  			skip();
106  			return hasNextByte();
107  		}
108  		public String next() {
109  			if (!hasNext())
110  				throw new NoSuchElementException();
111  			StringBuilder sb = new StringBuilder();
112  			int b = readByte();
113  			while (!isSpaceChar(b)) {
114  				sb.appendCodePoint(b);
115  				b = readByte();
116  			}
117  			return sb.toString();
118  		}
119  		public int nextInt() {
120  			if (!hasNext())
121  				throw new NoSuchElementException();
122  			int c = readByte();
123  			while (isSpaceChar(c))
124  				c = readByte();
125  			boolean minus = false;
126  			if (c == '-') {
127  				minus = true;
128  				c = readByte();
129  			}
130  			int res = 0;
131  			do {
132  				if (c < '0' || c > '9')
133  					throw new InputMismatchException();
134  				res = res * 10 + c - '0';
135  				c = readByte();
136  			} while (!isSpaceChar(c));
137  			return (minus) ? -res : res;
138  		}
139  		public long nextLong() {
140  			if (!hasNext())
141  				throw new NoSuchElementException();
142  			int c = readByte();
143  			while (isSpaceChar(c))
144  				c = readByte();
145  			boolean minus = false;
146  			if (c == '-') {
147  				minus = true;
148  				c = readByte();
149  			}
150  			long res = 0;
151  			do {
152  				if (c < '0' || c > '9')
153  					throw new InputMismatchException();
154  				res = res * 10 + c - '0';
155  				c = readByte();
156  			} while (!isSpaceChar(c));
157  			return (minus) ? -res : res;
158  		}
159  		public double nextDouble() {
160  			return Double.parseDouble(next());
161  		}
162  		public int[] nextIntArray(int n) {
163  			int[] a = new int[n];
164  			for (int i = 0; i < n; i++)
165  				a[i] = nextInt();
166  			return a;
167  		}
168  		public long[] nextLongArray(int n) {
169  			long[] a = new long[n];
170  			for (int i = 0; i < n; i++)
171  				a[i] = nextLong();
172  			return a;
173  		}
174  		public char[][] nextCharMap(int n, int m) {
175  			char[][] map = new char[n][m];
176  			for (int i = 0; i < n; i++)
177  				map[i] = next().toCharArray();
178  			return map;
179  		}
180  	}
181  	static void tr(Object... o) {
182  		out.println(Arrays.deepToString(o));
183  		out.flush();
184  	}
185  }
```


**Question**: Does `(x,27)` have data dependence over `(lsum,42)` in function `solve`? If so, provide a trace.

**Output**:


Ground truth answer: yes