[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**. 

```Python
1   from collections import defaultdict, deque
2   from heapq import heappush, heappop
3   from itertools import permutations, accumulate
4   import sys
5   import math
6   import bisect
7   def LI(): return [int(x) for x in sys.stdin.buffer.readline().split()]
8   def I(): return int(sys.stdin.buffer.readline())
9   def LS():return [list(x) for x in sys.stdin.readline().split()]
10  def S():
11      res = list(sys.stdin.readline())
12      if res[-1] == "\n":
13          return res[:-1]
14      return res
15  def IR(n):
16      return [I() for i in range(n)]
17  def LIR(n):
18      return [LI() for i in range(n)]
19  def SR(n):
20      return [S() for i in range(n)]
21  def LSR(n):
22      return [LS() for i in range(n)]
23  sys.setrecursionlimit(1000000)
24  def solve():
25      mod = 1000000007
26      s = S()
27      t = S()
28      ns = len(s)
29      nt = len(t)
30      f = {}
31      for i in range(26):
32          f[chr(i + ord("a"))] = i + 1
33      for i in range(ns):
34          s[i] = f[s[i]]
35      for i in range(nt):
36          t[i] = f[t[i]]
37      b = 10
38      mod1 = 1000000007
39      mod2 = 1000000009
40      pownt1 = pow(b,nt,mod)
41      invnt1 = pow(pownt1,mod1-2,mod1)
42      powns1 = pow(b,ns,mod1)
43      invns1 = pow(powns1,mod1-2,mod1)
44      pownt2 = pow(b,nt,mod2)
45      invnt2 = pow(pownt2,mod2-2,mod2)
46      powns2 = pow(b,ns,mod2)
47      invns2 = pow(powns2,mod2-2,mod2)
48      l = 0
49      r = ns+2
50      while r > l+1:
51          m = (l+r) >> 1
52          n = m*nt
53          st = [m]*nt
54          ms = n//ns
55          rs = n-ms*ns
56          ss = []
57          for i in range(ns):
58              if i < rs:
59                  ss.append(ms + 1)
60              else:
61                  ss.append(ms)
62          p1 = pow(b,n,mod1)
63          p2 = pow(b,n,mod2)
64          h1 = 0
65          h2 = 0
66          for i in range(nt):
67              ti = t[i]
68              pow1 = pow(b,(n-i)*mod1,mod1)
69              pows1 = pow(pownt1,st[i],mod1)
70              h1 += ti*pow1*(pows1-1)*pow(pows1*(1-invnt1),mod1-2,mod1)%mod1
71              h1 %= mod1
72              pow2 = pow(b,(n-i)*mod2,mod2)
73              pows2 = pow(pownt2,st[i],mod2)
74              h2 += ti*pow2*(pows2-1)*pow(pows2*(1-invnt2),mod2-2,mod2)%mod2
75              h2 %= mod2
76          k1 = 0
77          k2 = 0
78          for i in range(ns):
79              if not ss[i]:
80                  break
81              si = s[i]
82              pow1 = pow(b,(n-i)*mod1,mod1)
83              pows1 = pow(powns1,ss[i],mod1)
84              k1 += si*pow1*(pows1-1)*pow(pows1*(1-invns1),mod1-2,mod1)%mod1
85              k1 %= mod1
86              pow2 = pow(b,(n-i)*mod2,mod2)
87              pows2 = pow(powns2,ss[i],mod2)
88              k2 += si*pow2*(pows2-1)*pow(pows2*(1-invns2),mod2-2,mod2)%mod2
89              k2 %= mod2
90          k = 0
91          for j in range(rs,ns+rs):
92              if (h1,h2) == (k1,k2):
93                  l = m
94                  break
95              i = j%ns
96              si = s[i]
97              sk = s[k]
98              k1 -= sk*p1%mod1
99              k1 += si
100              k1 *= b
101              k1 %= mod1
102              k2 -= sk*p2%mod2
103              k2 += si
104              k2 *= b
105              k2 %= mod2
106              k += 1
107          else:
108              r = m
109      if l == ns+1:
110          print(-1)
111      else:
112          print(l)
113      return
114  if __name__ == "__main__":
115      solve()
```


**Question**: Does `(n,52)` have data dependence over `(rs,55)` in function `solve`? If so, provide a trace.

**Output**:


Ground truth answer: yes