[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.util.Arrays;
2   import java.util.Scanner;
3   public class Main {
4       Scanner   sc2   = new Scanner(System.in);
5       long      start = System.currentTimeMillis();
6       long      fin   = System.currentTimeMillis();
7       final int MOD   = 1000000007;
8       int[]     dx    = { 1, 0, 0, -1 };
9       int[]     dy    = { 0, 1, -1, 0 };
10      void run() {
11          MyScanner sc    = new MyScanner();
12          int n = sc.nextInt();
13          int m = sc.nextInt();
14          int M = 998244353;
15          if (m > 200000) {
16              return;
17          }
18          int[][] dp = new int[2][m];
19          int[] sum = new int[m];
20          dp[0][0] = 1;
21          dp[0][1] = 1;
22          sum[0] = 1;
23          sum[1] = 2;
24          for (int i = 2; i < m; i++) {
25              dp[0][i] = (dp[0][i - 1] + dp[0][i - 2]) % M;
26              sum[i] = (sum[i - 1] + dp[0][i]) % M;
27          }
28          long last = dp[0][m - 1];
29          for (int i = 1; i < n; i++) {
30              int[] next = new int[m];
31              next[0] = sum[0];
32              for (int j = 1; j < m; j++) 
33                  next[j] = (next[j - 1] + sum[j]) % M;
34              last = sum[m - 1];
35              sum = next;
36          }
37          System.out.println(last);
38      }
39      public static void main(String[] args) {
40          new Main().run();
41      }
42      void debug(Object... o) {
43          System.out.println(Arrays.deepToString(o));
44      }
45      void debug2(int[][] array) {
46          for (int i = 0; i < array.length; i++) {
47              for (int j = 0; j < array[i].length; j++) {
48                  System.out.print(array[i][j]);
49              }
50              System.out.println();
51          }
52      }
53      boolean inner(int h, int w, int limH, int limW) {
54          return 0 <= h && h < limH && 0 <= w && w < limW;
55      }
56      void swap(int[] x, int a, int b) {
57          int tmp = x[a];
58          x[a] = x[b];
59          x[b] = tmp;
60      }
61      int lower_bound(int a[], int k) {
62          int l = -1;
63          int r = a.length;
64          while (r - l > 1) {
65              int mid = (l + r) / 2;
66              if (k <= a[mid])
67                  r = mid;
68              else
69                  l = mid;
70          }
71          return r;
72      }
73      int upper_bound(int a[], int k) {
74          int l = -1;
75          int r = a.length;
76          while (r - l > 1) {
77              int mid = (l + r) / 2;
78              if (k < a[mid])
79                  r = mid;
80              else
81                  l = mid;
82          }
83          return r;
84      }
85      long gcd(long a, long b) {
86          return a % b == 0 ? b : gcd(b, a % b);
87      }
88      long lcm(long a, long b) {
89          return a * b / gcd(a, b);
90      }
91      boolean palindrome(String s) {
92          for (int i = 0; i < s.length() / 2; i++) {
93              if (s.charAt(i) != s.charAt(s.length() - 1 - i)) {
94                  return false;
95              }
96          }
97          return true;
98      }
99      class MyScanner {
100          int nextInt() {
101              try {
102                  int c = System.in.read();
103                  while (c != '-' && (c < '0' || '9' < c))
104                      c = System.in.read();
105                  if (c == '-')
106                      return -nextInt();
107                  int res = 0;
108                  do {
109                      res *= 10;
110                      res += c - '0';
111                      c = System.in.read();
112                  } while ('0' <= c && c <= '9');
113                  return res;
114              } catch (Exception e) {
115                  return -1;
116              }
117          }
118          double nextDouble() {
119              return Double.parseDouble(next());
120          }
121          long nextLong() {
122              return Long.parseLong(next());
123          }
124          String next() {
125              try {
126                  StringBuilder res = new StringBuilder("");
127                  int c = System.in.read();
128                  while (Character.isWhitespace(c))
129                      c = System.in.read();
130                  do {
131                      res.append((char) c);
132                  } while (!Character.isWhitespace(c = System.in.read()));
133                  return res.toString();
134              } catch (Exception e) {
135                  return null;
136              }
137          }
138          int[] nextIntArray(int n) {
139              int[] in = new int[n];
140              for (int i = 0; i < n; i++) {
141                  in[i] = nextInt();
142              }
143              return in;
144          }
145          int[][] nextInt2dArray(int n, int m) {
146              int[][] in = new int[n][m];
147              for (int i = 0; i < n; i++) {
148                  in[i] = nextIntArray(m);
149              }
150              return in;
151          }
152          double[] nextDoubleArray(int n) {
153              double[] in = new double[n];
154              for (int i = 0; i < n; i++) {
155                  in[i] = nextDouble();
156              }
157              return in;
158          }
159          long[] nextLongArray(int n) {
160              long[] in = new long[n];
161              for (int i = 0; i < n; i++) {
162                  in[i] = nextLong();
163              }
164              return in;
165          }
166          char[][] nextCharField(int n, int m) {
167              char[][] in = new char[n][m];
168              for (int i = 0; i < n; i++) {
169                  String s = sc.next();
170                  for (int j = 0; j < m; j++) {
171                      in[i][j] = s.charAt(j);
172                  }
173              }
174              return in;
175          }
176      }
177  }
```


**Question**: Does `(next,31)` have data dependence over `(sum,35)` in function `run`? If so, provide a trace.

**Output**:


Ground truth answer: yes