[INSTRUCTIONS]

You are a program-analysis assistant. Please perform a **static** analysis of “information flow” 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. Information Flow Definition

An **information flow** from variable `x` to variable `y`, written `x -> y`, exists whenever information stored in `x` is transferred to or used to derive information transferred to `y`. 

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


### Types of Direct Flows

- **Direct Explicit Flow**  
  A direct explicit flow occurs when the value written by one variable instance is directly used in computing the value of another, through operations like assignments.

  Formally, if `(varA, lineA)` writes a value and `(varB, lineB)` reads that value to compute its own, then there is a **direct explicit flow** from `(varA, lineA)` to `(varB, lineB)`, written as: `(varA,lineA) -> (varB,lineB)`. 


  **Example 1**:
    ```
    1  x = 5
    2  y = x + 1
    ```
    `(x,1) -> (y,2)` is a direct explicit flow.


  **Example 2**:
    ```
    1  x = 5
    2  if cond:
    3      y = x + 1
    ```
    `(x,1) -> (y,3)` is a direct explicit flow. 

    Under static analysis, we treat **each conditional or loop as potentially taking any outcome**, and we **do not perform semantic pruning**. Therefore, even if the statement at line 3 is conditionally executed, we still recognize `(x,1)` as used to compute `(y,3)`.
    


- **Direct Implicit Flow**

  A direct implicit flow from `(varA,lineA)` to `(varB, lineB)` occurs when `(varA,lineA)`’s value directly influences whether `lineB` executes. In other words, it corresponds to a **direct control dependence**, where the condition has **at least one branch** where `(varB, lineB)` must run and another where it may not. All conditional structures (e.g., if, while, for, switch, etc.) generate these flows.

  Formally, if `(varA,lineA)` is **directly read** in a condition at `lineC` and that condition **directly** decides whether `lineB` executes, then we say there is an **direct implicit flow** from `(varA,lineA)` to `(varB,lineB)` (or `(varA,lineA) -> (varB,lineB)`).

  **Example**:
  ```
  1  z = x + 5
  2  if z > 1:
  3      y = 10
  ```
  `(z,1) -> (z,2)* -> (y,3)` is a direct implicit flow through the condition at line 2. Line 2 is a conditional statement and the value of `(z,2)*` here is not redefined, therefore marked with "*".
  
  The asterisk * indicates that the variable is used for the conditional statement but not redefined at that line (e.g., used in a condition).



### Information Flow Between Variables

An **information flow** exists from a variable `(varA, lineA)` to another variable `(varB, lineB)` if there is a **transitive chain of direct flows** — explicit, implicit, or both — connecting them.

That is, there exists a sequence of variables:
`(varA, lineA) -> ... -> (varB, lineB)`



## 2. **Output Format**  

When asked "Which variable instances have information flow over `(targetVar, targetLine)`? List all such variables.", respond:, respond with a JSON object:

```json
   {
    "InfomationFlowSources": [
      ["valA", lineA],
      ["valB", lineB],
      ...
    ]
   }
   ```

If you believe there is no variables have information flow on `(targetVar, targetLine)`, respond:

```json
{
  "InfomationFlowSources": false
}
```


## 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  def example_func():
2      status = 0
3      flag = False
4      balance = 1000
5      balance += 500
6      if balance > 1000:
7          status = 1
8          flag = True
9      limit = status * 5000
10     transaction = limit * 0.2
11     return flag
```

#### Example Question 1.1:
Which variable instances have information flow over `(transaction,10)`? List all such variables.


**Analysis**:
- Line 10: `(transaction,10)` reads `limit`, so there is a direct explicit flow from `(limit,9)` to `(transaction,10)`. 
- Line 9: `(limit,9)` reads `status`, so there is a direct explicit flow from `(status,7)` (or `(status,2)`) to `(limit,9)`. 
- Line 7: `(status,7)` is updated only if line 6’s condition is true, so has a direct control dependence on line 6.
- Line 6: checks `balance > 1000`, reads from `(balance,5)`, so there is a direct implicit flow from `(balance,5)` to `(status,7)` through `balance` at line 6.
- Line 5: `(balance,5)` reads from `(balance,4)`, so there is a direct explicit flow from `(balance,4)` to `(balance,5)`.
- Line 4: `(balance,4)` initializes the variable balance.
  

**Output**:
```json
   {
    "InfomationFlowSources": [
      ["limit", 9],
      ["status", 2],
      ["status", 7],
      ["balance", 4],
      ["balance", 5],
    ]
   }
```

 
#### Example Question 1.2
Which variable instances have information flow over `(flag,3)`? List all such variables.

**Analysis**
Line 3 initializes `flag` without any data dependency or control dependency. 


**Output**:
```json
{
  "InfomationFlowSources": []
}
```



### Example 2
```python
1.  val = 5
2.  size = 3
3.  arr = [0] * size
4.  i = 0
5.  j = 1
6.  total = 0
7.  if val > 2:
8.      arr[j % size] = j
9.  while i < size:
10.     arr[j] += 1
11.     score = arr[j+1]
12.     total += score * 2
13.     diff = total - score
14.     j = (j + 1) % size
15.     i += 1
16. last = arr[-1]
17. summary = diff + j
```

#### Example Question 2.1:
Which variable instances have information flow over `(summary,17)`? List all such variables.

**Output**:
```json
   {
    "InfomationFlowSources": [
      ["diff", 13],
      ["j", 14],
      ["j", 5],
      ["size", 2],
      ["i", 4],
      ["i", 15],
    ]
   }
```


**Analysis**:
- Line 17: `(summary,17)` reads `diff` and `j`. `diff` is from line 13. Next, we focus on the path through `j`. `j` is updated from line 14 and line 5. 
- Static analysis assumption: Every branch or loop condition may succeed or fail; we do not prune unreachable paths. When line 9 never evaluates to `true`, `(summary,17)` reads `j` from `(j,5)`. When line 9 evaluates to `true`, line 14 will be executed. 
- Line 14: `(j,14)` is updated inside the `while i < size` loop, so its update is control-dependent on the loop condition at line 9. There is also an explicit flow from `size` to `j`.
- Line 9: `while i < size` reads `size` (a pure read), so has a direct explicit flow from `(size,2)`. It also reads `i` from line 15 or line 4.

**Output**:
```json
   {
    "InfomationFlowSources": [
      ["diff", 13],
      ["j", 14],
      ["j", 5],
      ["size", 2],
      ["i", 4],
      ["i", 15]
    ]
   }
```


#### Example Question 2.2:
Which variable instances have information flow over `(score,11)`? List all such variables.


**Analysis**:
- Line 11: `(score,11)` reads `arr[j+1]`. There is a direct data dependence on both `arr` and `j`. That `j` may come from either `(j,5)` (initial) or subsequent updates at `(j,14)`. `arr` may come from line 3, line 8, or line 10 given the index `j+1` is with an unknown value. 
- Static analysis assumption: Every branch or loop condition may succeed or fail; we do not prune unreachable paths. Therefore, `j` could be read from either line 5 (e.g., when it is the first iteration of the while loop) or line 14 (e.g., when it is the not the first iteration of the while loop). Similar for `arr`, it could be read from line 3, line 8, or line 10.
- If `arr` is read from line 10, there is an explicit flow from `(j,5)`, `(j,14)`, and `(size,2)` to `arr`. There is also an explicit flow from `arr` itself because the operation is `+=`, hence from `(arr,3)`, `(arr,8)`, or `(arr,10)`. There is also an implicit flow from `i` and `size` to line 10. Hence from `(i, 4)`, `(i,15)`, and `(size,2)`.
- If `arr` is read from line 8, there is an explicit flow from `(j,5)` and `(size,2)` to `arr`. There is also implicit flow from `val` (as in the condition in line 7) read from line 1.
- If `arr` is read from line 3, there is an explicit flow from `(size,2)` to `arr`.
- Line 11 is control-dependent on line 9, therefore there is a implicit flow from `i` (possibly from `(i,15)` or `(i,4)`) and `size` (possibly from `(size,2)`) to line 11.

**Output**:
```json
   {
    "InfomationFlowSources": [
      ["arr", 10],
      ["arr", 8],
      ["arr", 3],
      ["j", 14],
      ["j", 5],
      ["size", 2],
      ["i", 4],
      ["i", 15],
      ["val", 1]
    ]
   }
```



#### Example Question 2.3:
Which variable instances have information flow over `(last,16)`? List all such variables.

**Analysis**:
- Line 16: `(last,16)` reads `arr[-1]`. That element may have been modified by the assignment in line 8, by the loop updates in line 10, or by initialization at line 3. Under static analysis, multiple update sites may contribute. 
- If from Line 3: there is an explicit flow from `(size,2)` to it.
- If from line 8: `arr[j % size] = j` executes only if `val > 2`, so `(arr,8)` has a direct control dependence on `(val,1)` via the `if` at line 7. There is also an explicit flow from `(j,5)` and `(size,2)` to `arr`.
- If from line 10:  there is an explicit flow from `(j,5)`, `(j,14)`, and `(size,2)` to `arr`. There is also an explicit flow from `arr` itself because the operation is `+=`, hence from `(arr,3)`, `(arr,8)`, or `(arr,10)`. There is also an implicit flow from `i` and `size` to line 10. Hence from `(i, 4)`, `(i,15)`, and `(size,2)`.


**Output**:
```json
   {
    "InfomationFlowSources": [
      ["arr", 10],
      ["arr", 8],
      ["arr", 3],
      ["j", 14],
      ["j", 5],
      ["size", 2],
      ["i", 4],
      ["i", 15],
      ["val", 1]
    ]
   }
```

---

[YOUR TURN]

Below is **your target snippet**. 

```Java
1   import java.io.*;
2   import java.util.*;
3   public class Main {
4   	private static Scanner sc;
5   	private static Printer pr;
6   	private static void solve() {
7   		int h = sc.nextInt();
8   		int w = sc.nextInt();
9   		int[][] a = new int[h][w];
10  		for (int i = 0; i < h; i++) {
11  			for (int j = 0; j < w; j++) {
12  				a[i][j] = sc.nextInt();
13  			}
14  		}
15  		List<Integer> x = new ArrayList<>();
16  		List<Integer> y = new ArrayList<>();
17  		boolean flag = false;
18  		int prex = 0;
19  		int prey = 0;
20  		boolean odd;
21  		for (int i = 0; i < h; i++) {
22  			if (i % 2 == 1) {
23  				for (int j = w - 1; j >= 0; j--) {
24  					if (a[i][j] % 2 == 1) {
25  						odd = true;
26  					} else {
27  						odd = false;
28  					}
29  					if (odd && flag) {
30  						flag = !flag;
31  						x.add(prex);
32  						y.add(prey);
33  						x.add(j);
34  						y.add(i);
35  						prex = j;
36  						prey = i;
37  					} else if (odd && !flag) {
38  						flag = !flag;
39  						prex = j;
40  						prey = i;
41  					} else if (!odd && flag) {
42  						x.add(prex);
43  						y.add(prey);
44  						x.add(j);
45  						y.add(i);
46  						prex = j;
47  						prey = i;
48  					} else if (!odd && !flag) {
49  					}
50  				}
51  			} else {
52  				for (int j = 0; j < w; j++) {
53  					if (a[i][j] % 2 == 1) {
54  						odd = true;
55  					} else {
56  						odd = false;
57  					}
58  					if (odd && flag) {
59  						flag = !flag;
60  						x.add(prex);
61  						y.add(prey);
62  						x.add(j);
63  						y.add(i);
64  						prex = j;
65  						prey = i;
66  					} else if (odd && !flag) {
67  						flag = !flag;
68  						prex = j;
69  						prey = i;
70  					} else if (!odd && flag) {
71  						x.add(prex);
72  						y.add(prey);
73  						x.add(j);
74  						y.add(i);
75  						prex = j;
76  						prey = i;
77  					} else if (!odd && !flag) {
78  					}
79  				}
80  			}
81  		}
82  		pr.println(x.size() / 2);
83  		for (int i = 0, size = x.size(); i < size / 2 * 2; i++) {
84  			if (i % 2 == 0) {
85  				StringBuilder sb = new StringBuilder();
86  				sb.append(y.get(i) + 1);
87  				sb.append(' ');
88  				sb.append(x.get(i) + 1);
89  				sb.append(' ');
90  				pr.print(sb.toString());
91  			} else {
92  				StringBuilder sb = new StringBuilder();
93  				sb.append(y.get(i) + 1);
94  				sb.append(' ');
95  				sb.append(x.get(i) + 1);
96  				pr.println(sb.toString());
97  			}
98  		}
99  	}
100  	public static void main(String[] args) {
101  		sc = new Scanner(System.in);
102  		pr = new Printer(System.out);
103  		solve();
104  		pr.close();
105  		sc.close();
106  	}
107  	static class Scanner {
108  		BufferedReader br;
109  		Scanner (InputStream in) {
110  			br = new BufferedReader(new InputStreamReader(in));
111  		}
112  		private boolean isPrintable(int ch) {
113  			return ch >= '!' && ch <= '~';
114  		}
115  		private boolean isCRLF(int ch) {
116  			return ch == '\n' || ch == '\r' || ch == -1;
117  		}
118  		private int nextPrintable() {
119  			try {
120  				int ch;
121  				while (!isPrintable(ch = br.read())) {
122  					if (ch == -1) {
123  						throw new NoSuchElementException();
124  					}
125  				}
126  				return ch;
127  			} catch (IOException e) {
128  				throw new NoSuchElementException();
129  			}
130  		}
131  		String next() {
132  			try {
133  				int ch = nextPrintable();
134  				StringBuilder sb = new StringBuilder();
135  				do {
136  					sb.appendCodePoint(ch);
137  				} while (isPrintable(ch = br.read()));
138  				return sb.toString();
139  			} catch (IOException e) {
140  				throw new NoSuchElementException();
141  			}
142  		}
143  		int nextInt() {
144  			try {
145  				boolean negative = false;
146  				int res = 0;
147  				int limit = -Integer.MAX_VALUE;
148  				int radix = 10;
149  				int fc = nextPrintable();
150  				if (fc < '0') {
151  					if (fc == '-') {
152  						negative = true;
153  						limit = Integer.MIN_VALUE;
154  					} else if (fc != '+') {
155  						throw new NumberFormatException();
156  					}
157  					fc = br.read();
158  				}
159  				int multmin = limit / radix;
160  				int ch = fc;
161  				do {
162  					int digit = ch - '0';
163  					if (digit < 0 || digit >= radix) {
164  						throw new NumberFormatException();
165  					}
166  					if (res < multmin) {
167  						throw new NumberFormatException();
168  					}
169  					res *= radix;
170  					if (res < limit + digit) {
171  						throw new NumberFormatException();
172  					}
173  					res -= digit;
174  				} while (isPrintable(ch = br.read()));
175  				return negative ? res : -res;
176  			} catch (IOException e) {
177  				throw new NoSuchElementException();
178  			}
179  		}
180  		long nextLong() {
181  			try {
182  				boolean negative = false;
183  				long res = 0;
184  				long limit = -Long.MAX_VALUE;
185  				int radix = 10;
186  				int fc = nextPrintable();
187  				if (fc < '0') {
188  					if (fc == '-') {
189  						negative = true;
190  						limit = Long.MIN_VALUE;
191  					} else if (fc != '+') {
192  						throw new NumberFormatException();
193  					}
194  					fc = br.read();
195  				}
196  				long multmin = limit / radix;
197  				int ch = fc;
198  				do {
199  					int digit = ch - '0';
200  					if (digit < 0 || digit >= radix) {
201  						throw new NumberFormatException();
202  					}
203  					if (res < multmin) {
204  						throw new NumberFormatException();
205  					}
206  					res *= radix;
207  					if (res < limit + digit) {
208  						throw new NumberFormatException();
209  					}
210  					res -= digit;
211  				} while (isPrintable(ch = br.read()));
212  				return negative ? res : -res;
213  			} catch (IOException e) {
214  				throw new NoSuchElementException();
215  			}
216  		}
217  		float nextFloat() {
218  			return Float.parseFloat(next());
219  		}
220  		double nextDouble() {
221  			return Double.parseDouble(next());
222  		}
223  		String nextLine() {
224  			try {
225  				int ch;
226  				while (isCRLF(ch = br.read())) {
227  					if (ch == -1) {
228  						throw new NoSuchElementException();
229  					}
230  				}
231  				StringBuilder sb = new StringBuilder();
232  				do {
233  					sb.appendCodePoint(ch);
234  				} while (!isCRLF(ch = br.read()));
235  				return sb.toString();
236  			} catch (IOException e) {
237  				throw new NoSuchElementException();
238  			}
239  		}
240  		void close() {
241  			try {
242  				br.close();
243  			} catch (IOException e) {
244  			}
245  		}
246  	}
247  	static class Printer extends PrintWriter {
248  		Printer(OutputStream out) {
249  			super(out);
250  		}
251  	}
252  }
```


**Question**: Which variable instances have information flow over `(x,42)` in function `solve`? List all such variables.

**Output**:


Ground truth answer: yes