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import sys
from typing import List
from functools import reduce...

import sys
class Solution:
    # Optimized: O(1) Time, O(1) Space - Bitwise Trick...

import sys
from typing import List
class Solution:

import sys
from typing import List
class Solution:

import sys
# O(log N) Time, O(1) Space - Optimal
def is_palindrome_optimal(n: int) -> bool:...

import sys
# Optimal: O(log(min(A,B))) Time - Euclidean Algorithm (Recursive)
def gcd_recursive(a, b):...

import sys
def solve_digit_manipulation():
    try:...

import sys
def spiralTraversal_optimal(matrix):
    """O(N^2) Time, O(1) Space - Boundary Pointers"""...

import sys
from collections import defaultdict
def diagonalTraversal_optimal(matrix):...

import sys
def rotateMatrix_optimal(matrix):
    """O(N^2) Time, O(1) Space - Transpose then Reverse"""...

import sys
class Solution:
    # Optimized: O(N) Time, O(N) Space - Stack and Hash Map (Dictionary)...

import sys
class Solution:
    # Use paired lists or a tuple of tuples for the lookup table, ordered largest to smallest...

import sys
from collections import Counter
class Solution:...

import sys
class Solution:
    # O(N) Time, O(1) Space - Right-to-Left Traversal (Cleanest Optimal)...

import sys
def solve_valid_palindrome():
    try:

import sys
def solve_reverse_words():
    try:

import sys
def solve_compress_string():
    try:

import sys
def solve_vowels_consonants():
    try:

import sys
def solve_letter_coverage():
    try:

import sys
def solve_digit_string():
    try:

import java.util.Scanner;
class Solution {
    public static void solve_odd_even_index() {...

#include <iostream>
using namespace std;
void solve_half_diamond() {...

import sys
def solve_submatrix_sum():
    # Read N

import sys
def solve_lower_triangle_sum():
    # Read N...

import sys
def solve_matrix_sums():
    # Set up fast input reading...

import java.util.Scanner;
class Solution {
    public static long sumNaturalNumbers_optimal(long N) {...

#include <iostream>
using namespace std;
long long calculateFactorial_iterative(int N) {...

import java.util.Scanner;
class Solution {
    public static boolean isArithmeticProgression(int[] nums) {...

import java.util.Scanner;
class Solution {
    public static void performOperation(int A, int B, char op) {...

import java.util.Scanner;
import java.util.Arrays;
class Solution {...

import java.util.Scanner;
import java.util.Map;
import java.util.TreeMap; ...

import java.util.Scanner;
class Solution {
    public static int findDuplicate_optimal(int[] nums) {...

import java.util.Scanner;
class Solution {
    public static void calculateOddEvenSum_optimal(int N, Scanner scanner) {...

import java.util.Scanner;
class Solution {
    public static long findMissingNumber_summation(int[] nums) {...

import sys
def solve_max_element():
    try:

from collections import Counter
from typing import List
# --- Optimized Approach ---...

from typing import List
class Solution:
    # 1. Brute Force Approach: Time O(n^2), Space O(1)...

from typing import List
class Solution:
    # 1. Optimized Approach: Time O(n), Space O(n) - Dictionary/Hash Map...