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NFA to accept strings that has atleast one character occurring in a multiple of 3

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  • Last Updated : 21 Dec, 2021

Prerequisites: Finite Automata
Given a string str consisting of characters a, b and c, check if the number of occurrences of any character in the string is a multiple of 3 or not. 

Examples: 

Input: str = bc 
Output: ACCEPTED 
Explanation: The string consists 0 a’s and 3 * 0 = 0.

Input: str = abccc 
Output: ACCEPTED 
Explanation: The string consists 3 c’s. 

Input: str = abc 
Output: NOT ACCEPTED 
 

Approach: 
An NFA or a Nondeterministic Finite Automata is very similar to a DFA. It is a finite state machine that accepts a string(under some specific condition) if it reaches a final state, otherwise rejects it. The additional features which an NFA has are: 

  1. Null move is allowed i.e., it can move forward without reading symbols.
  2. Ability to transmit to any number of states for a particular input.

NFA Machine that accepts all strings in which the occurrences of atleast one character is a multiple of 3
For the above problem statement, we must first build an NFA machine. NFA machine is similar to a flowchart with various states and transitions. NFA machine corresponding to the above problem is shown below, Q3, Q4 and Q8 are the final states:
 

How does this NFA Machine work: 
The working of the machine depends on checking if the string has 3 multiples of a’s or b’s or c’s. 

  • Case 1: Number of a’s is a multiple of three:
    • To check whether the number of a’s in the string is a three multiple or not, a separate set of states is defined. The states defined as Q2, Q3, Q4 check whether the number of a’s is a multiple of three or not. If at any point this case reaches the final state Q2, then the number of a’s is a multiple of three.
  • Case 2: Number of b’s is a multiple of three:
    • To check whether the number of b’s in the string is a three multiple or not, a separate set of states is defined. The states defined as Q5, Q6, Q7 check whether the number of b’s is a multiple of three or not. If at any point this case reaches the final state Q5, then the number of b’s is a multiple of three.
  • Case 3: Number of c’s is a multiple of three:
    • To check whether the number of c’s in the string is a three multiple or not, a separate set of states is defined. The states defined as Q8, Q9, Q10 check whether the number of c’s is a multiple of three or not. If at any point this case reaches the final state Q8, then the number of c’s is a multiple of three.

Below is the implementation of the above approach:  

C++




// C++ implementation of the above approach
#include <bits/stdc++.h>
 
// NFA variable that keeps track of
// the state while transaction.
int nfa = 1;
 
// This checks for invalid input.
int flag = 0;
using namespace std;
 
// Function for the state Q2
void state1(char c)
{
    // State transitions
    // 'a' takes to Q4, and
    // 'b' and 'c' remain at Q2
    if (c == 'a')
        nfa = 2;
    else if (c == 'b' || c == 'c')
        nfa = 1;
    else
        flag = 1;
}
 
// Function for the state Q3
void state2(char c)
{
    // State transitions
    // 'a' takes to Q3, and
    // 'b' and 'c' remain at Q4
    if (c == 'a')
        nfa = 3;
    else if (c == 'b' || c == 'c')
        nfa = 2;
    else
        flag = 1;
}
 
// Function for the state Q4
void state3(char c)
{
    // State transitions
    // 'a' takes to Q2, and
    // 'b' and 'c' remain at Q3
    if (c == 'a')
        nfa = 1;
    else if (c == 'b' || c == 'c')
        nfa = 3;
    else
        flag = 1;
}
 
// Function for the state Q5
void state4(char c)
{
    // State transitions
    // 'b' takes to Q6, and
    // 'a' and 'c' remain at Q5
    if (c == 'b')
        nfa = 5;
    else if (c == 'a' || c == 'c')
        nfa = 4;
    else
        flag = 1;
}
 
// Function for the state Q6
void state5(char c)
{
    // State transitions
    // 'b' takes to Q7, and
    // 'a' and 'c' remain at Q7
    if (c == 'b')
        nfa = 6;
    else if (c == 'a' || c == 'c')
        nfa = 5;
    else
        flag = 1;
}
 
// Function for the state Q7
void state6(char c)
{
    // State transitions
    // 'b' takes to Q5, and
    // 'a' and 'c' remain at Q7
    if (c == 'b')
        nfa = 4;
    else if (c == 'a' || c == 'c')
        nfa = 6;
    else
        flag = 1;
}
 
// Function for the state Q8
void state7(char c)
{
    // State transitions
    // 'c' takes to Q9, and
    // 'a' and 'b' remain at Q8
    if (c == 'c')
        nfa = 8;
    else if (c == 'b' || c == 'a')
        nfa = 7;
    else
        flag = 1;
}
 
// Function for the state Q9
void state8(char c)
{
    // State transitions
    // 'c' takes to Q10, and
    // 'a' and 'b' remain at Q9
    if (c == 'c')
        nfa = 9;
    else if (c == 'b' || c == 'a')
        nfa = 8;
    else
        flag = 1;
}
 
// Function for the state Q10
void state9(char c)
{
    // State transitions
    // 'c' takes to Q8, and
    // 'a' and 'b' remain at Q10
    if (c == 'c')
        nfa = 7;
    else if (c == 'b' || c == 'a')
        nfa = 9;
    else
        flag = 1;
}
 
// Function to check for 3 a's
bool checkA(string s, int x)
{
    for (int i = 0; i < x; i++) {
        if (nfa == 1)
            state1(s[i]);
        else if (nfa == 2)
            state2(s[i]);
        else if (nfa == 3)
            state3(s[i]);
    }
    if (nfa == 1) {
        return true;
    }
    else {
        nfa = 4;
    }
}
 
// Function to check for 3 b's
bool checkB(string s, int x)
{
    for (int i = 0; i < x; i++) {
        if (nfa == 4)
            state4(s[i]);
        else if (nfa == 5)
            state5(s[i]);
        else if (nfa == 6)
            state6(s[i]);
    }
    if (nfa == 4) {
 
        return true;
    }
    else {
        nfa = 7;
    }
}
 
// Function to check for 3 c's
bool checkC(string s, int x)
{
    for (int i = 0; i < x; i++) {
        if (nfa == 7)
            state7(s[i]);
        else if (nfa == 8)
            state8(s[i]);
        else if (nfa == 9)
            state9(s[i]);
    }
    if (nfa == 7) {
 
        return true;
    }
}
 
// Driver Code
int main()
{
    string s = "bbbca";
    int x = 5;
 
    // If any of the states is true, that is, if either
    // the number of a's or number of b's or number of c's
    // is a multiple of three, then the string is accepted
    if (checkA(s, x) || checkB(s, x) || checkC(s, x)) {
        cout << "ACCEPTED";
    }
 
    else {
        if (flag == 0) {
            cout << "NOT ACCEPTED";
            return 0;
        }
        else {
            cout << "INPUT OUT OF DICTIONARY.";
            return 0;
        }
    }
}

Java




// Java implementation of the above approach
class GFG {
     
// NFA variable that keeps track of
// the state while transaction.
static int nfa = 1;
 
// This checks for invalid input.
static int flag = 0;
 
// Function for the state Q2
static void state1(char c)
{
    // State transitions
    // 'a' takes to Q4, and
    // 'b' and 'c' remain at Q2
    if (c == 'a')
        nfa = 2;
    else if (c == 'b' || c == 'c')
        nfa = 1;
    else
        flag = 1;
}
 
// Function for the state Q3
static void state2(char c)
{
    // State transitions
    // 'a' takes to Q3, and
    // 'b' and 'c' remain at Q4
    if (c == 'a')
        nfa = 3;
    else if (c == 'b' || c == 'c')
        nfa = 2;
    else
        flag = 1;
}
 
// Function for the state Q4
static void state3(char c)
{
    // State transitions
    // 'a' takes to Q2, and
    // 'b' and 'c' remain at Q3
    if (c == 'a')
        nfa = 1;
    else if (c == 'b' || c == 'c')
        nfa = 3;
    else
        flag = 1;
}
 
// Function for the state Q5
static void state4(char c)
{
    // State transitions
    // 'b' takes to Q6, and
    // 'a' and 'c' remain at Q5
    if (c == 'b')
        nfa = 5;
    else if (c == 'a' || c == 'c')
        nfa = 4;
    else
        flag = 1;
}
 
// Function for the state Q6
static void state5(char c)
{
    // State transitions
    // 'b' takes to Q7, and
    // 'a' and 'c' remain at Q7
    if (c == 'b')
        nfa = 6;
    else if (c == 'a' || c == 'c')
        nfa = 5;
    else
        flag = 1;
}
 
// Function for the state Q7
static void state6(char c)
{
    // State transitions
    // 'b' takes to Q5, and
    // 'a' and 'c' remain at Q7
    if (c == 'b')
        nfa = 4;
    else if (c == 'a' || c == 'c')
        nfa = 6;
    else
        flag = 1;
}
 
// Function for the state Q8
static void state7(char c)
{
    // State transitions
    // 'c' takes to Q9, and
    // 'a' and 'b' remain at Q8
    if (c == 'c')
        nfa = 8;
    else if (c == 'b' || c == 'a')
        nfa = 7;
    else
        flag = 1;
}
 
// Function for the state Q9
static void state8(char c)
{
    // State transitions
    // 'c' takes to Q10, and
    // 'a' and 'b' remain at Q9
    if (c == 'c')
        nfa = 9;
    else if (c == 'b' || c == 'a')
        nfa = 8;
    else
        flag = 1;
}
 
// Function for the state Q10
static void state9(char c)
{
    // State transitions
    // 'c' takes to Q8, and
    // 'a' and 'b' remain at Q10
    if (c == 'c')
        nfa = 7;
    else if (c == 'b' || c == 'a')
        nfa = 9;
    else
        flag = 1;
}
 
// Function to check for 3 a's
static boolean checkA(String s, int x)
{
    for (int i = 0; i < x; i++) {
        if (nfa == 1)
            state1(s.charAt(i));
        else if (nfa == 2)
            state2(s.charAt(i));
        else if (nfa == 3)
            state3(s.charAt(i));
    }
    if (nfa == 1) {
        return true;
    }
    else {
        nfa = 4;
    }
    return false;
}
 
// Function to check for 3 b's
static boolean checkB(String s, int x)
{
    for (int i = 0; i < x; i++) {
        if (nfa == 4)
            state4(s.charAt(i));
        else if (nfa == 5)
            state5(s.charAt(i));
        else if (nfa == 6)
            state6(s.charAt(i));
    }
    if (nfa == 4) {
 
        return true;
    }
    else {
        nfa = 7;
    }
    return false;
}
 
// Function to check for 3 c's
static boolean checkC(String s, int x)
{
    for (int i = 0; i < x; i++) {
        if (nfa == 7)
            state7(s.charAt(i));
        else if (nfa == 8)
            state8(s.charAt(i));
        else if (nfa == 9)
            state9(s.charAt(i));
    }
    if (nfa == 7) {
 
        return true;
    }
    return false;
}
 
// Driver Code
public static void main (String[] args)
{
    String s = "bbbca";
    int x = 5;
 
    // If any of the states is true, that is, if either
    // the number of a's or number of b's or number of c's
    // is a multiple of three, then the string is accepted
    if (checkA(s, x) || checkB(s, x) || checkC(s, x)) {
        System.out.println("ACCEPTED");
    }
 
    else {
        if (flag == 0) {
            System.out.println("NOT ACCEPTED");
             
        }
        else {
            System.out.println("INPUT OUT OF DICTIONARY.");
             
        }
    }
}
}
 
// This code is contributed by AnkitRai01

Python3




# Python3 implementation of the above approach
 
# NFA variable that keeps track of
# the state while transaction.
nfa = 1
 
# This checks for invalid input.
flag = 0
 
# Function for the state Q2
def state1(c):
    global nfa,flag
 
    # State transitions
    # 'a' takes to Q4, and
    # 'b' and 'c' remain at Q2
    if (c == 'a'):
        nfa = 2
    elif (c == 'b' or c == 'c'):
        nfa = 1
    else:
        flag = 1
 
# Function for the state Q3
def state2(c):
    global nfa,flag
 
    # State transitions
    # 'a' takes to Q3, and
    # 'b' and 'c' remain at Q4
    if (c == 'a'):
        nfa = 3
    elif (c == 'b' or c == 'c'):
        nfa = 2
    else:
        flag = 1
 
# Function for the state Q4
def state3(c):
    global nfa,flag
 
    # State transitions
    # 'a' takes to Q2, and
    # 'b' and 'c' remain at Q3
    if (c == 'a'):
        nfa = 1
    elif (c == 'b' or c == 'c'):
        nfa = 3
    else:
        flag = 1
 
# Function for the state Q5
def state4(c):
    global nfa,flag
 
    # State transitions
    # 'b' takes to Q6, and
    # 'a' and 'c' remain at Q5
    if (c == 'b'):
        nfa = 5
    elif (c == 'a' or c == 'c'):
        nfa = 4
    else:
        flag = 1
 
# Function for the state Q6
def state5(c):
    global nfa, flag
 
    # State transitions
    # 'b' takes to Q7, and
    # 'a' and 'c' remain at Q7
    if (c == 'b'):
        nfa = 6
    elif (c == 'a' or c == 'c'):
        nfa = 5
    else:
        flag = 1
 
# Function for the state Q7
def state6(c):
    global nfa,flag
 
    # State transitions
    # 'b' takes to Q5, and
    # 'a' and 'c' remain at Q7
    if (c == 'b'):
        nfa = 4
    elif (c == 'a' or c == 'c'):
        nfa = 6
    else:
        flag = 1
 
# Function for the state Q8
def state7(c):
    global nfa,flag
 
    # State transitions
    # 'c' takes to Q9, and
    # 'a' and 'b' remain at Q8
    if (c == 'c'):
        nfa = 8
    elif (c == 'b' or c == 'a'):
        nfa = 7
    else:
        flag = 1
 
# Function for the state Q9
def state8(c):
    global nfa,flag
 
    # State transitions
    # 'c' takes to Q10, and
    # 'a' and 'b' remain at Q9
    if (c == 'c'):
        nfa = 9
    elif (c == 'b' or c == 'a'):
        nfa = 8
    else:
        flag = 1
 
# Function for the state Q10
def state9(c):
    global nfa,flag
 
    # State transitions
    # 'c' takes to Q8, and
    # 'a' and 'b' remain at Q10
    if (c == 'c'):
        nfa = 7
    elif (c == 'b' or c == 'a'):
        nfa = 9
    else:
        flag = 1
         
# Function to check for 3 a's
def checkA(s, x):
    global nfa,flag
    for i in range(x):
        if (nfa == 1):
            state1(s[i])
        elif (nfa == 2):
            state2(s[i])
        elif (nfa == 3):
            state3(s[i])
     
    if (nfa == 1):
        return True
     
    else:
        nfa = 4
     
# Function to check for 3 b's
def checkB(s, x):
    global nfa,flag
    for i in range(x):
        if (nfa == 4):
            state4(s[i])
        elif (nfa == 5):
            state5(s[i])
        elif (nfa == 6):
            state6(s[i])
     
    if (nfa == 4):
        return True
    else:
        nfa = 7
     
# Function to check for 3 c's
def checkC(s, x):
    global nfa, flag
    for i in range(x):
        if (nfa == 7):
            state7(s[i])
        elif (nfa == 8):
            state8(s[i])
        elif (nfa == 9):
            state9(s[i])
             
    if (nfa == 7):
        return True
 
# Driver Code
 
s = "bbbca"
x = 5
 
# If any of the states is True, that is, if either
# the number of a's or number of b's or number of c's
# is a multiple of three, then the is accepted
if (checkA(s, x) or checkB(s, x) or checkC(s, x)):
    print("ACCEPTED")
 
else:
    if (flag == 0):
        print("NOT ACCEPTED")
     
    else:
        print("INPUT OUT OF DICTIONARY.")
         
# This code is contributed by shubhamsingh10

C#




// C# implementation of the above approach
using System;
 
class GFG {
      
// NFA variable that keeps track of
// the state while transaction.
static int nfa = 1;
  
// This checks for invalid input.
static int flag = 0;
  
// Function for the state Q2
static void state1(char c)
{
    // State transitions
    // 'a' takes to Q4, and
    // 'b' and 'c' remain at Q2
    if (c == 'a')
        nfa = 2;
    else if (c == 'b' || c == 'c')
        nfa = 1;
    else
        flag = 1;
}
  
// Function for the state Q3
static void state2(char c)
{
    // State transitions
    // 'a' takes to Q3, and
    // 'b' and 'c' remain at Q4
    if (c == 'a')
        nfa = 3;
    else if (c == 'b' || c == 'c')
        nfa = 2;
    else
        flag = 1;
}
  
// Function for the state Q4
static void state3(char c)
{
    // State transitions
    // 'a' takes to Q2, and
    // 'b' and 'c' remain at Q3
    if (c == 'a')
        nfa = 1;
    else if (c == 'b' || c == 'c')
        nfa = 3;
    else
        flag = 1;
}
  
// Function for the state Q5
static void state4(char c)
{
    // State transitions
    // 'b' takes to Q6, and
    // 'a' and 'c' remain at Q5
    if (c == 'b')
        nfa = 5;
    else if (c == 'a' || c == 'c')
        nfa = 4;
    else
        flag = 1;
}
  
// Function for the state Q6
static void state5(char c)
{
    // State transitions
    // 'b' takes to Q7, and
    // 'a' and 'c' remain at Q7
    if (c == 'b')
        nfa = 6;
    else if (c == 'a' || c == 'c')
        nfa = 5;
    else
        flag = 1;
}
  
// Function for the state Q7
static void state6(char c)
{
    // State transitions
    // 'b' takes to Q5, and
    // 'a' and 'c' remain at Q7
    if (c == 'b')
        nfa = 4;
    else if (c == 'a' || c == 'c')
        nfa = 6;
    else
        flag = 1;
}
  
// Function for the state Q8
static void state7(char c)
{
    // State transitions
    // 'c' takes to Q9, and
    // 'a' and 'b' remain at Q8
    if (c == 'c')
        nfa = 8;
    else if (c == 'b' || c == 'a')
        nfa = 7;
    else
        flag = 1;
}
  
// Function for the state Q9
static void state8(char c)
{
    // State transitions
    // 'c' takes to Q10, and
    // 'a' and 'b' remain at Q9
    if (c == 'c')
        nfa = 9;
    else if (c == 'b' || c == 'a')
        nfa = 8;
    else
        flag = 1;
}
  
// Function for the state Q10
static void state9(char c)
{
    // State transitions
    // 'c' takes to Q8, and
    // 'a' and 'b' remain at Q10
    if (c == 'c')
        nfa = 7;
    else if (c == 'b' || c == 'a')
        nfa = 9;
    else
        flag = 1;
}
  
// Function to check for 3 a's
static bool checkA(String s, int x)
{
    for (int i = 0; i < x; i++) {
        if (nfa == 1)
            state1(s[i]);
        else if (nfa == 2)
            state2(s[i]);
        else if (nfa == 3)
            state3(s[i]);
    }
    if (nfa == 1) {
        return true;
    }
    else {
        nfa = 4;
    }
    return false;
}
  
// Function to check for 3 b's
static bool checkB(String s, int x)
{
    for (int i = 0; i < x; i++) {
        if (nfa == 4)
            state4(s[i]);
        else if (nfa == 5)
            state5(s[i]);
        else if (nfa == 6)
            state6(s[i]);
    }
    if (nfa == 4) {
  
        return true;
    }
    else {
        nfa = 7;
    }
    return false;
}
  
// Function to check for 3 c's
static bool checkC(String s, int x)
{
    for (int i = 0; i < x; i++) {
        if (nfa == 7)
            state7(s[i]);
        else if (nfa == 8)
            state8(s[i]);
        else if (nfa == 9)
            state9(s[i]);
    }
    if (nfa == 7) {
  
        return true;
    }
    return false;
}
  
// Driver Code
public static void Main(String[] args)
{
    String s = "bbbca";
    int x = 5;
  
    // If any of the states is true, that is, if either
    // the number of a's or number of b's or number of c's
    // is a multiple of three, then the string is accepted
    if (checkA(s, x) || checkB(s, x) || checkC(s, x)) {
        Console.WriteLine("ACCEPTED");
    }
  
    else {
        if (flag == 0) {
            Console.WriteLine("NOT ACCEPTED");
              
        }
        else {
            Console.WriteLine("INPUT OUT OF DICTIONARY.");
              
        }
    }
}
}
 
// This code is contributed by 29AjayKumar

Javascript




<script>
 
// JavaScript implementation of the above approach
 
// NFA variable that keeps track of
// the state while transaction.
let nfa = 1;
 
// This checks for invalid input.
let flag = 0;
 
// Function for the state Q2
function state1(c) {
    // State transitions
    // 'a' takes to Q4, and
    // 'b' and 'c' remain at Q2
    if (c == 'a')
        nfa = 2;
    else if (c == 'b' || c == 'c')
        nfa = 1;
    else
        flag = 1;
}
 
// Function for the state Q3
function state2(c) {
    // State transitions
    // 'a' takes to Q3, and
    // 'b' and 'c' remain at Q4
    if (c == 'a')
        nfa = 3;
    else if (c == 'b' || c == 'c')
        nfa = 2;
    else
        flag = 1;
}
 
// Function for the state Q4
function state3(c) {
    // State transitions
    // 'a' takes to Q2, and
    // 'b' and 'c' remain at Q3
    if (c == 'a')
        nfa = 1;
    else if (c == 'b' || c == 'c')
        nfa = 3;
    else
        flag = 1;
}
 
// Function for the state Q5
function state4(c) {
    // State transitions
    // 'b' takes to Q6, and
    // 'a' and 'c' remain at Q5
    if (c == 'b')
        nfa = 5;
    else if (c == 'a' || c == 'c')
        nfa = 4;
    else
        flag = 1;
}
 
// Function for the state Q6
function state5(c) {
    // State transitions
    // 'b' takes to Q7, and
    // 'a' and 'c' remain at Q7
    if (c == 'b')
        nfa = 6;
    else if (c == 'a' || c == 'c')
        nfa = 5;
    else
        flag = 1;
}
 
// Function for the state Q7
function state6(c) {
    // State transitions
    // 'b' takes to Q5, and
    // 'a' and 'c' remain at Q7
    if (c == 'b')
        nfa = 4;
    else if (c == 'a' || c == 'c')
        nfa = 6;
    else
        flag = 1;
}
 
// Function for the state Q8
function state7(c) {
    // State transitions
    // 'c' takes to Q9, and
    // 'a' and 'b' remain at Q8
    if (c == 'c')
        nfa = 8;
    else if (c == 'b' || c == 'a')
        nfa = 7;
    else
        flag = 1;
}
 
// Function for the state Q9
function state8(c) {
    // State transitions
    // 'c' takes to Q10, and
    // 'a' and 'b' remain at Q9
    if (c == 'c')
        nfa = 9;
    else if (c == 'b' || c == 'a')
        nfa = 8;
    else
        flag = 1;
}
 
// Function for the state Q10
function state9(c) {
    // State transitions
    // 'c' takes to Q8, and
    // 'a' and 'b' remain at Q10
    if (c == 'c')
        nfa = 7;
    else if (c == 'b' || c == 'a')
        nfa = 9;
    else
        flag = 1;
}
 
// Function to check for 3 a's
function checkA(s, x) {
    for (let i = 0; i < x; i++) {
        if (nfa == 1)
            state1(s[i]);
        else if (nfa == 2)
            state2(s[i]);
        else if (nfa == 3)
            state3(s[i]);
    }
    if (nfa == 1) {
        return true;
    }
    else {
        nfa = 4;
    }
}
 
// Function to check for 3 b's
function checkB(s, x) {
    for (let i = 0; i < x; i++) {
        if (nfa == 4)
            state4(s[i]);
        else if (nfa == 5)
            state5(s[i]);
        else if (nfa == 6)
            state6(s[i]);
    }
    if (nfa == 4) {
 
        return true;
    }
    else {
        nfa = 7;
    }
}
 
// Function to check for 3 c's
function checkC(s, x) {
    for (let i = 0; i < x; i++) {
        if (nfa == 7)
            state7(s[i]);
        else if (nfa == 8)
            state8(s[i]);
        else if (nfa == 9)
            state9(s[i]);
    }
    if (nfa == 7) {
 
        return true;
    }
}
 
// Driver Code
 
let s = "bbbca";
let x = 5;
 
// If any of the states is true, that is, if either
// the number of a's or number of b's or number of c's
// is a multiple of three, then the string is accepted
if (checkA(s, x) || checkB(s, x) || checkC(s, x)) {
    document.write("ACCEPTED");
}
 
else {
    if (flag == 0) {
        document.write("NOT ACCEPTED");
    }
    else {
        document.write("INPUT OUT OF DICTIONARY.");
    }
}
 
// This code is contributed by gfgking
 
</script>

Output: 

ACCEPTED

 

Time Complexity: O(x)

Auxiliary Space: O(1)


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