[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 "Which variable instances have data dependence over `(targetVar, targetLine)`? List all such variables.", respond:
   ```json
   {
    "DataDependenceSources": [
      ["valA", lineA],
      ["valB", lineB],
      ...
    ]
   }
   ```

  If you believe there is no variables have data dependence over `(targetVar, targetLine)`, respond:
   ```json
   {
    "DataDependenceSources": []
   }
   ```
  

## 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:
Which variable instances have data dependence over `(step,9)`? List all such variables.


**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
   {
    "DataDependenceSources": [
      ["step", 9],
      ["step", 3],
      ["value", 2],
    ]
   }
```





#### Example Question 1.2 
Which variable instances have data dependence over `(total,7)`? List all such variables.


**Analysis**:
- Line 7: `total += value`, meaning `(total,7)` reads from `total` at line 1 or 7 (itself), and `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)`



**Output**:
```json
   {
    "DataDependenceSources": [
      ["total", 1],
      ["total", 7],
      ["value", 5],
      ["value", 2]
    ]
   }
```



### 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:
Which variable instances have data dependence over `(arr,6)`? List all such variables.



**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)`.




**Output**:
```json
   {
    "DataDependenceSources": [
      ["arr", 6],
      ["arr", 4],
      ["arr", 1],
      ["i", 5],
      ["x", 2]
    ]
   }
```


#### Example Question 2.2:
Which variable instances have data dependence over `(temp,7)`? List all such variables.


**Analysis**:
- Line 7: `temp = arr[i]`, meaning temp depends on `i`, since `i` determines which element of arr is accessed. And `arr[i]` can only refer to `(arr,6)` as they have the same index.
- Line 5: `i` is initialized and iterates over the range 0 to 3.


**Output**:
```json
   {
    "DataDependenceSources": [
      ["arr", 6],
      ["i", 5],
    ]
   }
```

#### Example Question 2.3:
Which variable instances have data dependence over `(result,9)`? List all such variables.

**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).
  - If it is from line 6: `arr[i] += 1`, `arr[i]` is modified based on the value of `i`. Line 5: `i` iterates over 0 to 3.
  - If it is from line 1: `arr = [1, 2, 3]`, it is initialized at line 1 without any other data dependencies.

**Output**:
```json
   {
    "DataDependenceSources": [
      ["arr", 6],
      ["arr", 1],
      ["i", 5],
    ]
   }
```

---

[YOUR TURN]

Below is **your target snippet**. 

```Python
1   from sys import stdout
2   from bisect import bisect_left as binl
3   from copy import copy, deepcopy
4   mod = 1
5   def intin():
6       input_tuple = input().split()
7       if len(input_tuple) <= 1:
8           return int(input_tuple[0])
9       return tuple(map(int, input_tuple))
10  def intina():
11      return [int(i) for i in input().split()]
12  def intinl(count):
13      return [intin() for _ in range(count)]
14  def modadd(x, y):
15      global mod
16      return (x + y) % mod
17  def modmlt(x, y):
18      global mod
19      return (x * y) % mod
20  def lcm(x, y):
21      while y != 0:
22          z = x % y
23          x = y
24          y = z
25      return x
26  def combination(x, y):
27      assert(x >= y)
28      if y > x // 2:
29          y = x - y
30      ret = 1
31      for i in range(0, y):
32          j = x - i
33          i = i + 1
34          ret = ret * j
35          ret = ret // i
36      return ret
37  def get_divisors(x):
38      retlist = []
39      for i in range(1, int(x**0.5) + 3):
40          if x % i == 0:
41              retlist.append(i)
42              retlist.append(x // i)
43      return retlist
44  def get_factors(x):
45      retlist = []
46      for i in range(2, int(x**0.5) + 3):
47          while x % i == 0:
48              retlist.append(i)
49              x = x // i
50      retlist.append(x)
51      return retlist
52  def make_linklist(xylist):
53      linklist = {}
54      for a, b in xylist:
55          linklist.setdefault(a, [])
56          linklist.setdefault(b, [])
57          linklist[a].append(b)
58          linklist[b].append(a)
59      return linklist
60  def calc_longest_distance(linklist, v=1):
61      distance_list = {}
62      distance_count = 0
63      distance = 0
64      vlist_previous = []
65      vlist = [v]
66      nodecount = len(linklist)
67      while distance_count < nodecount:
68          vlist_next = []
69          for v in vlist:
70              distance_list[v] = distance
71              distance_count += 1
72              vlist_next.extend(linklist[v])
73          distance += 1
74          vlist_to_del = vlist_previous
75          vlist_previous = vlist
76          vlist = list(set(vlist_next) - set(vlist_to_del))
77      max_distance = -1
78      max_v = None
79      for v, distance in distance_list.items():
80          if distance > max_distance:
81              max_distance = distance
82              max_v = v
83      return (max_distance, max_v)
84  def calc_tree_diameter(linklist, v=1):
85      _, u = calc_longest_distance(linklist, v)
86      distance, _ = calc_longest_distance(linklist, u)
87      return distance
88  def main():
89      a, b = intin()
90      if a % 2 == 1 and b % 2 == 1:
91          print('Yes')
92      else:
93          print('No')
94  if __name__ == '__main__':
95      main()
```


**Question**: Which variable instances have data dependence over `(vlist_previous,75)` in function `calc_longest_distance`? List all such variables.

**Output**:


Ground truth answer: yes