[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 __future__ import print_function
2   import sys
3   import time
4   sys.setrecursionlimit(10 ** 6)
5   inputFileName = None
6   outputFileName = None
7   if inputFileName is not None:
8       outputFileName = inputFileName[:-3] + ".out"
9   import platform
10  global MY_DEBUG
11  MY_DEBUG = platform.node().lower() in map(lambda s: s.lower(), ['HOMEPC3', 'lenovo'])
12  logF = None
13  if MY_DEBUG:
14      logF = open("run.log", 'w')
15  def dbg_print(s):
16      if MY_DEBUG:
17          logF.write('{0}\n'.format(s))
18          logF.flush()
19  startTime = time.time()
20  dbg_print(startTime)
21  def runSingleTest(inpF, outF):
22      line = inpF.readline()
23      tokens = line.split()
24      a_exc = int(tokens[0])
25      b_inc = int(tokens[1])
26      line = inpF.readline()
27      n = int(line)
28      for i in xrange(n):
29          guess = (a_exc + b_inc + 1) / 2
30          print(guess, file=outF)
31          line = inpF.readline_safe()
32          if 'CORRECT' == line:
33              return
34          elif 'TOO_SMALL' == line:
35              a_exc = guess + 1
36          elif 'TOO_BIG' == line:
37              b_inc = guess - 1
38          else:
39              dbg_print("Unexpected line '{0}'".format(line))
40              raise RuntimeError("Unexpected line '{0}'".format(line))
41      raise RuntimeError("AAAAA")
42  class InpFWrapper:
43      def __init__(self, inpF):
44          self.inpF = inpF
45      def readline(self):
46          line = self.inpF.readline()
47          dbg_print("Read line is '{0}'".format(line.replace("\n", "\\n")))
48          return line
49      def readline_safe(self):
50          line = self.inpF.readline()
51          dbg_print("Read line safe is '{0}'".format(line.replace("\n", "\\n")))
52          return line[0:-1] if (line[-1] == '\n') else line
53  class OutFWrapper:
54      def __init__(self, outF):
55          self.outF = outF
56      def __getattr__(self, name):
57          return getattr(self.outF, name)
58      def write(self, str):
59          dbg_print("Writing '{0}'".format(str.replace("\n", "\\n")))
60          self.outF.write(str)
61          self.outF.flush()
62  def myMain(inpF, outF):
63      if (MY_DEBUG):
64          dbg_print('input = {0}, out = {1}'.format(inputFileName, outputFileName))
65      inpF = InpFWrapper(inpF)
66      outF = OutFWrapper(outF)
67      line = inpF.readline()
68      testsCount = int(line)
69      for i in xrange(1, testsCount + 1):
70          dbg_print('--------  {0}/{1} {2} --------------------------'.format(i, testsCount, (time.time() - startTime)))
71          runSingleTest(inpF, outF)
72          dbg_print('--------  {0}/{1} {2} --------------------------'.format(i, testsCount, (time.time() - startTime)))
73  if inputFileName is not None:
74      with open(inputFileName) as inpF:
75          with open(outputFileName, 'w') as outF:
76              myMain(inpF, outF)
77  else:
78      myMain(sys.stdin, sys.stdout)
79  dbg_print("Finished!!!! Total time = {0}".format((time.time() - startTime)))
```


**Question**: Does `(tokens,23)` have data dependence over `(n,27)` in function `runSingleTest`? If so, provide a trace.

**Output**:


Ground truth answer: no