[INSTRUCTIONS]

You are a program-analysis assistant. Your task is to statically analyze the control dependence of a given code snippet. 

## 1. Control Dependence Definition
  Control dependence captures the influence of control-flow decisions on the execution of statements.

  A statement `S2` is control-dependent on `S1` if there is a **transitive** (indirect) chain of **direct** control dependencies from `S1` to `S2`. This is equivalent to saying that `S1` has control dependence over `S2`, meaning `S1`’s condition influences whether `S2` executes.


  **Direct Control Dependence**:
  A statement `S2` is **directly** control-dependent on a statement `S1` if:
  
  1.	`S1` is a conditional control statement (e.g., an `if`, `while`, `for`, `switch`, etc.).
  
  2.	`S1` directly determines whether `S2` executes. That is, `S1` has multiple successor branches, 

  - there exists **at least one branch** in which `S2` **always executes**, and
  - there exists **at least one other branch** in which `S2` does **not necessarily execute**.

    “Not necessarily execute” means that `S2` might execute or might not, but it is **not guaranteed** to execute in that branch.

  `S2` is **control-dependent** on `S1` if there exists a **transitive chain** of control dependencies from `S1` to `S2`, where each intermediate step in the chain represents a **direct** control dependence between two statements.



## 2. Output Format
When asked, “Does line `S1` have control dependence over line `S2`? If so, provide a trace.” You should respond in JSON format as follows:

- If there is a control dependence:

```json
{
  "ControlDependence": true,
  "Trace": [S1, ..., S2]
}
```

- If there is no control dependence:
```json
{
  "ControlDependence": false
}
```

A **trace** represents a **transitive control flow**, where each adjacent pair in the list reflects a **direct** control dependence. The trace must starts with `S1` and ends with `S2`.


    

## 3. Interprocedural Control 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  if x > 0:
2      y = 10       # this line is directly control-dependent on line 1 (x>0)
3      if y > 5:
4          z = 20   # this line is directly control-dependent on line 3 (y>5)
5          w = 30   # this line is directly control-dependent on line 3 (y>5)
6  v = 40           # this line is NOT control-dependent on any line
```


**Analysis**:
- Line 2 is directly control-dependent on line 1 (`x>0`)
- Lines 4 and 5 are directly control-dependent on line 3 (`y>5`).
- Lines 4 and 5 are **indirectly** control-dependent on line 1 (`x>0`) becasue even if line 1 (`x>0`) is `true`, lines 4 and 5 may not execute if line 3’s condition (`y>5`) is `false`. However, since control dependence is transitive, lines 4 and 5 are **indirectly** control-dependent on line 1. 
- Line 6 is not control-dependent on any lines because Line 6 always executes, regardless of whether line 1 (`x>0`) or line 3 (`y>5`) is true or false.



#### Example Question 1.1: 
Does line `1` have control dependence over line `5`? If so, provide a trace.

**Output**: 
```json
{
  "ControlDependence": true,
  "Trace": [1, 3, 5]
}
```

#### Example Question 1.2:
Does line `3` have control dependence over line `6`? If so, provide a trace.

**Output**: 
```json
{
  "ControlDependence": false
}
```


### Example 2

```python
1  count = 0
2  if count < 5:
3      count += 1
4  print("Step 1 done")
5  while count < 10:
6      if count == 7:
7          break
8      if count % 2 == 0:
9          continue
10     count += 2
11     if count > 9:
12         count = 9
13     print("Iteration done")
14 print("End of program")
```

#### Example Question 2.1: 
Does line `5` have control dependence over line `13`? If so, provide a trace.

**Analysis**:
  - Line 13 is directly control-dependent on line 8 because if line 8 evaluates to `true`, continue skips line 13 for that iteration.
  - Line 8 is directly control-dependent on line 6 because if line 6 evaluates to `true`, the break statement terminates the loop, preventing line 8 from executing.
  - Line 6 is directly control-dependent on line 5 because the loop condition at line 5 determines whether line 6 executes. If line 5 evaluates to `false`, execution skips the loop entirely.


Since control dependence is transitive, line 5 indirectly controls line 13 through lines 6 and 8. The trace is `5->6->8->13`.

**Output**:
```json
{
  "ControlDependence": true,
  "Trace": [5, 6, 8, 13]
}
```

#### Example Question 2.2: 
Does line `8` have control dependence over line `10`? If so, provide a trace.


**Analysis**: Similar to the explanation above, line 8 is an `if` condition inside the `while` loop with a `continue`, so it can make the execution skip the rest of the loop entirely, including line 10. Therefore, line 10 directly control-dependent on 8.



**Output**:
```json
{
  "ControlDependence": true,
  "Trace": [8, 10]
}
```

#### Example Question 2.3: 
Does line `5` have control dependence over line `14`? If so, provide a trace.

**Analysis**: Line 5 doesn't affect whether line 14 is reached. Line 14 is outside the while loop (lines 5 -- 13) and will be executed regardless of the condition. 

**Output**:
```json
{
  "ControlDependence": false
}
```


#### Example Question 2.4: 
Does line `2` have control dependence over line `12`? If so, provide a trace.

**Analysis**: Line 2 doesn't affect whether line 12 is reached. The condition in line 2 only controls whether line 3 is executed. Line 12 is in while loop (line 5 -- 13) and will or will not be executed regardless of the condition in line 2. 

**Output**:
```json
{
  "ControlDependence": false
}
```

---

[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 line `31` have control dependence over line `84` in function `solve`? If so, provide a trace.

**Output**:


Ground truth answer: no