[INSTRUCTIONS]

You are a program-analysis assistant. Please perform a **static** analysis of “information flow” 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. Information Flow Definition

An **information flow** from variable `x` to variable `y`, written `x -> y`, exists whenever information stored in `x` is transferred to or used to derive information transferred to `y`. 

We denote each variable instance as `(var,lineNumber)`, meaning the variable `var` is **defined or updated** at `lineNumber`.


### Types of Direct Flows

- **Direct Explicit Flow**  
  A direct explicit flow occurs when the value written by one variable instance is directly used in computing the value of another, through operations like assignments.

  Formally, if `(varA, lineA)` writes a value and `(varB, lineB)` reads that value to compute its own, then there is a **direct explicit flow** from `(varA, lineA)` to `(varB, lineB)`, written as: `(varA,lineA) -> (varB,lineB)`. 


  **Example 1**:
    ```
    1  x = 5
    2  y = x + 1
    ```
    `(x,1) -> (y,2)` is a direct explicit flow.


  **Example 2**:
    ```
    1  x = 5
    2  if cond:
    3      y = x + 1
    ```
    `(x,1) -> (y,3)` is a direct explicit flow. 

    Under static analysis, we treat **each conditional or loop as potentially taking any outcome**, and we **do not perform semantic pruning**. Therefore, even if the statement at line 3 is conditionally executed, we still recognize `(x,1)` as used to compute `(y,3)`.
    


- **Direct Implicit Flow**

  A direct implicit flow from `(varA,lineA)` to `(varB, lineB)` occurs when `(varA,lineA)`’s value directly influences whether `lineB` executes. In other words, it corresponds to a **direct control dependence**, where the condition has **at least one branch** where `(varB, lineB)` must run and another where it may not. All conditional structures (e.g., if, while, for, switch, etc.) generate these flows.

  Formally, if `(varA,lineA)` is **directly read** in a condition at `lineC` and that condition **directly** decides whether `lineB` executes, then we say there is an **direct implicit flow** from `(varA,lineA)` to `(varB,lineB)` (or `(varA,lineA) -> (varB,lineB)`).

  **Example**:
  ```
  1  z = x + 5
  2  if z > 1:
  3      y = 10
  ```
  `(z,1) -> (z,2)* -> (y,3)` is a direct implicit flow through the condition at line 2. Line 2 is a conditional statement and the value of `(z,2)*` here is not redefined, therefore marked with "*".
  
  The asterisk * indicates that the variable is used for the conditional statement but not redefined at that line (e.g., used in a condition).



### Information Flow Between Variables

An **information flow** exists from a variable `(varA, lineA)` to another variable `(varB, lineB)` if there is a **transitive chain of direct flows** — explicit, implicit, or both — connecting them.

That is, there exists a sequence of variables:
`(varA, lineA) -> ... -> (varB, lineB)`



## 2. **Output Format**  

When asked "Is there information flow from `(varA,lineA)` to `(varB,lineB)`? If so, provide one feasible trace.", respond with a JSON object:

- If information flow exists:
```json
{
  "InformationFlow": true,
  "Trace": [
    {
      "from": ["varOrExpr", lineNumber            /*, "use" optional */],
      "to":   ["varOrExpr", lineNumber            /*, "use" optional */],
      "type": "data" | "control"
    },
    // … additional edges
  ]
}
```
- If no information flow exists, omit the Trace field:
```json
{
  "InformationFlow": false
}
```

- Every trace edge must specify "from", "to" (each may include a third "use" element when the line is a conditional statement that only reads without redefining/updating the value), and "type".
- **We do not consider loop in the trace**.


A **trace** represents a **transitive chain of direct flows** arise from compositions of **direct** implicit or explicit flows, where each edge represents a **direct implicit flow** (through control dependence) or a **direct explicit flow** edge (through data dependence).


You only need to provide **one** valid trace if you conclude a information flow 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  def example_func():
2      status = 0
3      flag = False
4      balance = 1000
5      balance += 500
6      if balance > 1000:
7          status = 1
8          flag = True
9      limit = status * 5000
10     transaction = limit * 0.2
11     return flag
```

#### Example Question 1.1:
Is there information flow from `(balance,4)` to `(transaction,10)`? If so, provide a trace.

**Analysis**:
- Line 10: `(transaction,10)` reads `limit`, so there is a direct explicit flow from `(limit,9)` to `(transaction,10)`. 
- Line 9: `(limit,9)` reads `status`, so there is a direct explicit flow from `(status,7)` (or `(status,2)`) to `(limit,9)`. Here we focus on `(status,7)`. 
- Line 7: `(status,7)` is updated only if line 6’s condition is true, so has a direct control dependence on line 6.
- Line 6: checks `balance > 1000`, reads from `(balance,5)`, so there is a direct implicit flow from `(balance,5)` to `(status,7)` through `balance` at line 6.
- Line 5: `(balance,5)` reads from `(balance,4)`, so there is a direct explicit flow from `(balance,4)` to `(balance,5)`.
- Line 4: `(balance,4)` initializes the variable balance.
  
Hence, A transitive of flow of direct implicit or explicit flows forms a information flow trace.


**Output**:
```json
{
  "InformationFlow": true,
  "Trace": [
    {
      "from": ["balance", 4],
      "to":   ["balance", 5],
      "type": "data"
    },
    {
      "from": ["balance", 5],
      "to":   ["balance", 6, "use"],
      "type": "data"
    },
    {
      "from": ["balance", 6, "use"],
      "to":   ["status", 7],
      "type": "control"
    },
    {
      "from": ["status", 7],
      "to":   ["limit", 9],
      "type": "data"
    },
    {
      "from": ["limit", 9],
      "to":   ["transaction", 10],
      "type": "data"
    }
  ]
}
```

 
- The edge from line 6 to `(status,7)` is through control dependence and therefore with a type of "control". 
- Line 6 only reads the value of `balance` without defining/updating it. So it is marked with `"use"`. 
- You only need to provide **one** valid chain if you conclude a the information flow exists. 


#### Example Question 1.2

Is there information flow from  `(flag,3)` to `(limit,9)`? If so, provide a trace.

**Analysis**
- Line 9: `(limit,9)` reads `status`, so there is a direct explicit flow from either `(status,7)` (if line 6’s condiation evaluates to `true`) or on the original `(status,2)` (if not) to `(limit,9)`. 
- Since this is a static analysis, all possible paths are considered without analyzing actual reachability. This means line 6 is treated as being evaluated to both possibly true and possibly false.
  - If line 6 is true, `(limit,9)` depends on `(status,7)`. 
    - Line 7: `(status,7)` doesn't have any data dependence as it is assigned with a constant number. The execution of line 7 is control-dependent on the condition in line 6.
    - Line 6: the condition at line 6 directly reads from `(balance,5)`. 
    - Line 5: `(balance,5)` is updated based on the previous value of `balance` in line 4.
    - Line 4: `(balance,4)` is initialized with a constant number. Therefore it has no more data dependence. Nor is it guarded by any conditions.
  - If line 6 is not true, `(limit,9)` depends on `(status,2)`. 
    - Line 2: `(status,2)` is initialized with a constant number. Therefore it has no more data dependence. Nor is it guarded by any conditions.
- Therefore, among all the possible paths, there is no transitive (indirect) chain of implicit or explicit flow from `(flag,3)` to `(limit,9)`. The information flow doesn't exist.


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

#### Example Question 1.3

Is there information flow from  `(status,2)` to `(transaction,10)`? If so, provide a trace.

**Analysis**
- Line 10: `(transaction,10)` reads `limit`, so there is a direct explicit flow from `(limit,9)` to `(transaction,10)`.
- Line 9: `(limit,9)` reads `status`, o there is a direct explicit flow from either `(status,7)`, or the original `(status,2)`, to `(limit,9)`.
- Similar to the above example, line 6 is possibly evaluated to true or false.
- If line 6 evaluates to `true`, there is a direct explicit flow from `(status,7)` to `(limit,9)`, which directly data-dependent on `(status,2)`, forming a transitive explicit flow trace :`(status,2) -> (status,7) -> (limit,9) -> (transaction,10)`.
- If line 6 evaluates to `false`, `(limit,9)` has a direct data dependence on `(status,2)`, forming a transitive explicit flow trace :`(status,2) -> (limit,9) -> (transaction,10)`.
- A transitive of explicit flow forms a information flow trace. Only one valid trace is required for the output even if multiple exists.

**Output**:
```json
{
  "InformationFlow": true,
  "Trace": [
    {
      "from": ["status", 2],
      "to":   ["limit", 9],
      "type": "data"
    },
    {
      "from": ["limit", 9],
      "to":   ["transaction", 10],
      "type": "data"
    }
  ]
}
```


### Example 2
```python
1.  val = 5
2.  size = 3
3.  arr = [0] * size
4.  i = 0
5.  j = 1
6.  total = 0
7.  if val > 2:
8.      arr[j % size] = j
9.  while i < size:
10.     arr[j] += 1
11.     score = arr[j+1]
12.     total += score * 2
13.     diff = total - score
14.     j = (j + 1) % size
15.     i += 1
16. last = arr[-1]
17. summary = diff + j
```

#### Example Question 2.1:
Is there information flow from  `(size,2)` to `(summary,17)`? If so, provide a trace.

**Analysis**:
- Line 17: `(summary,17)` reads `diff` and `j`. We focus on the path through `j`. `j` is updated from line 14 and line 5. 
- Static analysis assumption: Every branch or loop condition may succeed or fail; we do not prune unreachable paths. When line 9 evaluates to `true`, line 14 will be executed. 
- Line 14: `(j,14)` is updated inside the `while i < size` loop, so its update is control-dependent on the loop condition at line 9.
- Line 9: `while i < size` reads `size` (a pure read), so has a direct explicit flow from `(size,2)`.
- Line 2: `size` is initialized with a constant number. It is not guarded with any conditions. 
- Therefore, there is an transitive trace of direct implicit/excplit flows from `(size,2)` to `(summary,17)`.

Only one valid trace is required for the output. 

**Output**:
```json
{
  "InformationFlow": true,
  "Trace": [
    {
      "from": ["size", 2],
      "to":   ["size", 9, "use"],
      "type": "data"
    },
    {
      "from": ["size", 9, "use"],
      "to":   ["j", 14],
      "type": "control"
    },
    {
      "from": ["j", 14],
      "to":   ["summary", 17],
      "type": "data"
    }
  ]
}
```


#### Example Question 2.2:
Is there information flow from  `(j,5)` to `(score,11)`? If so, provide a trace. 

**Analysis**:
- Line 11: `(score,11)` reads `arr[j+1]`. There is a direct data dependence on both `arr` and `j`. That `j` may come from either `(j,5)` (initial) or subsequent updates at `(j,14)`.
- Static analysis assumption: Every branch or loop condition may succeed or fail; we do not prune unreachable paths. Therefore, `j` could be read from either line 5 (e.g., when it is the first iteration of the while loop) or line 14 (e.g., when it is the not the first iteration of the while loop).
- If line 11 reads `j` from line 5, it forms a tansitive explicit flow trace: `(j,5) -> (score,11)`.
- If line 11 reads `j` from line 14, it forms a tansitive explicit flow trace: `(j,5) -> (j,14) -> (score,11)`.
- A transitive of explicit flow forms a information flow trace. Only one valid trace is required for the output even if multiple exists.


**Output**:
```json
{
  "InformationFlow": true,
  "Trace": [
    {
      "from": ["j", 5],
      "to":   ["score", 11],
      "type": "data"
    }
  ]
}
```



#### Example Question 2.3:
Is there information flow from  `(j,1)` to `(i,15)`? If so, provide a trace.

**Analysis**: 
- Line 15: `(i,15)` updates `i` based on its previous value. So there is a direct explicit flow from either `(i,4)` (initialization) or itself (loop-carried) to `(i,15)`. But we do not consider loop-carried dependencies across iterations in this analysis. So we focus on `(i,4)`.
- The execution of line 15 is conditional on the while loop at line 9.
- Line 9: `while i < size` depends on `i` and `size`. It directly reads from `(i,4)` (or possibly `(i,15)`) and `(size,2)`.
- Both `(i,4)` and `(size,2)` are initialized with constants and are not themselves control-dependent on any earlier values or branches.
- There is no use of `(j,1)` in the computation of `(i,15)`, nor does `(j,1)` influence whether line 15 executes.
- Therefore, there is no transitive flow, explicit or implicit, from `(j,1)` to `(i,15)`.


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


#### Example Question 2.4:
Is there information flow from  `(val,1)` to `(last,16)`? If so, provide a trace.

**Analysis**:
- Line 16: `(last,16)` reads `arr[-1]`. That element may have been modified by the assignment in line 8 (or by the loop updates in line 10). Under static analysis, multiple update sites may contribute. We conservatively pick one valid trace (line 8).
- Line 8: `arr[j % size] = j` executes only if `val > 2`, so `(arr,8)` has a direct control dependence on `(val,1)` via the `if` at line 7.
- Therefore, there exists a trace from `(val,1)` to `(last,16)`, and the edge from line 7 to `(arr,8)` is through implicit and explicit flows. 

**Output**:
```json
{
  "InformationFlow": true,
  "Trace": [
    {
      "from": ["val", 1],
      "to":   ["val", 7, "use"],
      "type": "data"
    },
    {
      "from": ["val", 7, "use"],
      "to":   ["arr", 8],
      "type": "control"
    },
    {
      "from": ["arr", 8],
      "to":   ["last", 16],
      "type": "data"
    }
  ]
}
```

---

[YOUR TURN]

Below is **your target snippet**. 

```Java
1   import java.io.IOException;
2   import java.io.InputStream;
3   import java.util.Arrays;
4   import java.util.NoSuchElementException;
5   import java.util.Random;
6   import java.util.Scanner;
7   class UnionFind{
8   	int Parent[];
9   	UnionFind(int n){
10  		Parent=new int[n];
11  		Arrays.fill(Parent, -1);
12  	}
13  	int root(int A) {
14  		if(Parent[A]<0)return A;
15  		return Parent[A]=root(Parent[A]);
16  	}
17  	int size(int A) {
18  		return -Parent[root(A)];
19  	}
20  	boolean connect(int A,int B) {
21  		A=root(A);
22  		B=root(B);
23  		if(A==B) return false;
24  		if(size(A)<size(B)) {int C=0;C=B;B=A;A=C;}
25  		Parent[A]+=Parent[B];
26  		Parent[B]=A;
27  		return true;
28  	}
29  }
30  public class Main {
31  	static FastScanner scan=new FastScanner();
32  	static Scanner scanner=new Scanner(System.in);
33  	static Random rand=new Random();
34  	static long mod=1000000007;
35  	static double eps=0.0000000001;
36  	static int big=Integer.MAX_VALUE;
37  	static long modlcm(long a,long b) {return a*b*modint(gcd(a,b),mod);}
38  	static long gcd (long a, long b) {return b>0?gcd(b,a%b):a;}
39  	static long lcm (long a, long b) {return a*b/gcd(a,b);}
40  	static int max(int a,int b) {return a>b?a:b;}
41  	static int min(int a,int b) {return a<b?a:b;}
42  	static long lmax(long a,long b) {return Math.max(a, b);}
43  	static long lmin(long a,long b) {return Math.min(a, b);}
44  	static long factorial(int i) {return i==1?1:i*factorial(i-1);}
45  	public static void main(String[] args) throws IOException {
46  		String str=scan.next();
47  		boolean b=false;
48  		for(int i=0;i<str.length();i++) {
49  			if(str.charAt(i)=='C')b=true;
50  			if(b&&str.charAt(i)=='F') {
51  				System.out.println("Yes");
52  				return;
53  			}
54  		}
55  		System.out.println("No");
56  	}
57  	static int lower_bound(int a[],int key) {
58  		int right=a.length;
59  		int left=0;
60  		while(right-left>1) {
61  			int mid=(right+left)/2;
62  			if(a[mid]<key) {
63  				left=mid;
64  			}
65  			else {
66  				right=mid;
67  			}
68  		}
69  		return left;
70  	}
71  	static int upper_bound(int a[],int key) {
72  		int right=a.length;
73  		int left=0;
74  		while(right-left>1) {
75  			int mid=(right+left)/2;
76  			if(a[mid]<=key) {
77  				left=mid;
78  			}
79  			else {
80  				right=mid;
81  			}
82  		}
83  		return left;
84  	}
85  	static boolean isPrime (long n) {
86  		if (n==2) return true;
87  		if (n<2 || n%2==0) return false;
88  		double d = Math.sqrt(n);
89  		for (int i=3; i<=d; i+=2)if(n%i==0){return false;}
90  		return true;
91  	}
92  	static int upper_division(int a,int b) {
93  		if(a%b==0) {
94  			return a/b;
95  		}
96  		else {
97  			return a/b+1;
98  		}
99  	}
100  	static long lupper_division(long a,long b) {
101  		if(a%b==0) {
102  			return a/b;
103  		}
104  		else {
105  			return a/b+1;
106  		}
107  	}
108  	static int[] setArray(int a) {
109  		int b[]=new int[a];
110  		for(int i=0;i<a;i++) {
111  			b[i]=scan.nextInt();
112  		}
113  		return b;
114  	}
115  	static long[] lsetArray(int a) {
116  		long b[]=new long[a];
117  		for(int i=0;i<a;i++) {
118  			b[i]=scan.nextLong();
119  		}
120  		return b;
121  	}
122  	static String reverce(String str) {
123  		String strr="";
124  		for(int i=str.length()-1;i>=0;i--) {
125  			strr+=str.charAt(i);
126  		}
127  		return strr;
128  	}
129  	public static void printArray(int[] que) {
130  		for(int i=0;i<que.length-1;i++) {
131  			System.out.print(que[i]+" ");
132  		}
133  		System.out.println(que[que.length-1]);
134  	}
135  	public static int[][] doublesort(int[][]a) {
136  		Arrays.sort(a,(x,y)->Integer.compare(x[0],y[0]));
137  		return a;
138  	}
139  	public static long[][] ldoublesort(long[][]a) {
140  		Arrays.sort(a,(x,y)->Long.compare(x[0],y[0]));
141  		return a;
142  	}
143  	static long modpow(long x,long n,long mo) {
144  		long sum=1;
145  		while(n>0) {
146  			if((n&1)==1) {
147  				sum=sum*x%mo;
148  			}
149  			x=x*x%mo;
150  			n>>=1;
151  		}
152  		return sum;
153  	}
154  	public static char[] revch(char ch[]) {
155  		char ret[]=new char[ch.length];
156  		for(int i=ch.length-1,j=0;i>=0;i--,j++) {
157  			ret[j]=ch[i];
158  		}
159  		return ret;
160  	}
161  	public static int[] revint(int ch[]) {
162  		int ret[]=new int[ch.length];
163  		for(int i=ch.length-1,j=0;i>=0;i--,j++) {
164  			ret[j]=ch[i];
165  		}
166  		return ret;
167  	}
168  	public static void warshall_floyd(int v[][],int n) {
169  		for(int k=0;k<n;k++)
170  			for(int i=0;i<n;i++)
171  				for(int j=0;j<n;j++)
172  					v[i][j]=min(v[i][j],v[i][k]+v[k][j]);
173  	}
174  	public static long modint(long a,long m) {
175  		long b=m,u=1,v=0;
176  		while(b!=0) {
177  			long t=a/b;
178  			a-=t*b;
179  			long x=a;
180  			a=b;
181  			b=x;
182  			u-=t*v;
183  			x=u;
184  			u=v;
185  			v=x;
186  		}
187  		u%=m;
188  		if(u<0)u+=m;
189  		return u;
190  	}
191  }
192  class FastScanner {
193  	private final InputStream in = System.in;
194  	private final byte[] buffer = new byte[1024];
195  	private int ptr = 0;
196  	private int buflen = 0;
197  	private boolean hasNextByte() {
198  		if (ptr < buflen) {
199  			return true;
200  		}else{
201  			ptr = 0;
202  			try {
203  				buflen = in.read(buffer);
204  			} catch (IOException e) {
205  				e.printStackTrace();
206  			}
207  			if (buflen <= 0) {
208  				return false;
209  			}
210  		}
211  		return true;
212  	}
213  	private int readByte() { if (hasNextByte()) return buffer[ptr++]; else return -1;}
214  	private static boolean isPrintableChar(int c) { return 33 <= c && c <= 126;}
215  	public boolean hasNext() { while(hasNextByte() && !isPrintableChar(buffer[ptr])) ptr++; return hasNextByte();}
216  	public String next() {
217  		if (!hasNext()) throw new NoSuchElementException();
218  		StringBuilder sb = new StringBuilder();
219  		int b = readByte();
220  		while(isPrintableChar(b)) {
221  			sb.appendCodePoint(b);
222  			b = readByte();
223  		}
224  		return sb.toString();
225  	}
226  	public long nextLong() {
227  		if (!hasNext()) throw new NoSuchElementException();
228  		long n = 0;
229  		boolean minus = false;
230  		int b = readByte();
231  		if (b == '-') {
232  			minus = true;
233  			b = readByte();
234  		}
235  		if (b < '0' || '9' < b) {
236  			throw new NumberFormatException();
237  		}
238  		while(true){
239  			if ('0' <= b && b <= '9') {
240  				n *= 10;
241  				n += b - '0';
242  			}else if(b == -1 || !isPrintableChar(b)){
243  				return minus ? -n : n;
244  			}else{
245  				throw new NumberFormatException();
246  			}
247  			b = readByte();
248  		}
249  	}
250  	public int nextInt() {
251  		long nl = nextLong();
252  		if (nl < Integer.MIN_VALUE || nl > Integer.MAX_VALUE) throw new NumberFormatException();
253  		return (int) nl;
254  	}
255  	public double nextDouble() { return Double.parseDouble(next());}
256  	public char nextchar() {
257  		try { return (char)System.in.read(); } catch(Exception e) {
258  			throw new RuntimeException(e);
259  		}
260  	}
261  }
```


**Question**: Is there information flow from `(b,247)` to `(n,241)` in function `nextLong`? If so, provide a trace.

**Output**:


Ground truth answer: yes