[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   import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,random,time,copy,functools
2   sys.setrecursionlimit(10**7)
3   def LI(): return [int(x) for x in sys.stdin.readline().split()]
4   def LI_(): return [int(x)-1 for x in sys.stdin.readline().split()]
5   def LF(): return [float(x) for x in sys.stdin.readline().split()]
6   def LS(): return sys.stdin.readline().split()
7   def I(): return int(sys.stdin.readline())
8   def F(): return float(sys.stdin.readline())
9   def S(): return input()
10  def pf(s): return print(s, flush=True)
11  def main():
12      inf = 10**20
13      eps = 1.0 / 10**10
14      mod = 10**9+7
15      ddn = [(-1,0),(-1,1),(0,1),(1,1),(1,0),(1,-1),(0,-1),(-1,-1)]
16      dd = [(-1,0),(0,1),(1,0),(0,-1)]
17      rr = []
18      while True:
19          n = I()
20          if n == 0:
21              break
22          a = []
23          for _ in range(n):
24              a.append(LS())
25          d = collections.defaultdict(int)
26          ed = collections.defaultdict(lambda: None)
27          for i in range(n):
28              x, y, di = a[i]
29              x = int(x)
30              y = int(y)
31              d[(x,y)] = i + 1
32              if di == 'x':
33                  d[(x+1,y)] = i + 1
34                  ed[(x,y)] = (x+1,y)
35                  ed[(x+1,y)] = (x,y)
36              else:
37                  d[(x,y+1)] = i + 1
38                  ed[(x,y)] = (x,y+1)
39                  ed[(x,y+1)] = (x,y)
40          ee = collections.defaultdict(set)
41          dka = list(d.keys())
42          for x,y in list(d.keys()):
43              dt = d[(x,y)]
44              for di,dj in dd:
45                  ni = x + di
46                  nj = y + dj
47                  if d[(ni,nj)] > 0 and d[(ni,nj)] != dt:
48                      ee[(x,y)].add((ni,nj))
49          v = collections.defaultdict(bool)
50          f = True
51          for k in dka:
52              if v[k]:
53                  continue
54              s1 = set()
55              s2 = set()
56              ns1 = set([k])
57              ns2 = set()
58              while ns1 or ns2:
59                  s1 |= ns1
60                  na = list(ns1)
61                  ns1 = set()
62                  for k in na:
63                      ns1 |= ee[k]
64                      ns2.add(ed[k])
65                  ns2 -= s2
66                  na = list(ns2)
67                  s2 |= ns2
68                  ns2 = set()
69                  for k in na:
70                      ns2 |= ee[k]
71                      ns1.add(ed[k])
72                  ns1 -= s1
73                  ns2 -= s2
74                  s1 |= ns1
75                  s2 |= ns2
76                  if s1 & s2:
77                      f = False
78                      break
79              if f:
80                  for k in s1:
81                      v[k] = 1
82                  for k in s2:
83                      v[k] = 1
84              else:
85                  break
86          if f:
87              rr.append('Yes')
88          else:
89              rr.append('No')
90      return '\n'.join(map(str,rr))
91  print(main())
```


**Question**: Does line `51` have control dependence over line `81` in function `main`? If so, provide a trace.

**Output**:


Ground truth answer: yes