Your task is to write a C$++$ program that reads multiple binary expressions with two duodecimal operands and a single integer operator from standard input. For each expression, the program should:

Validate the expression for correct duodecimal operands and a valid operator.

Convert the duodecimal operands to their decimal equivalents.

Compute the result of the operation in decimal, handling division by zero and other potential errors gracefully.

Convert the decimal result back to a duodecimal representation.

Display the result in duodecimal form or indicate if the expression is invalid.

Expressions that are considered invalid include those with characters other than $0-9,$ A$,$ B$,$ a$,$ b for the operands or with invalid operators.

The Duodecimal System Explained

The duodecimal system, also known as base$-12$ or dozenal, is a positional notation numeral system using twelve as its base. It extends the common decimal numeral system by two extra digits to accommodate twelve distinct digit symbols. In this system, the numbers after $9$ are represented by the letters $\text{'}$A$\text{'}($which represents $10$ in decimal$)$ and $\text{'}$B$\text{'}($which represents $11$ in decimal$).$

Converting from Duodecimal to Decimal

To convert a number from duodecimal to decimal:

Steps:

Initialize the resultant decimal number to $0.$

Traverse the duodecimal string from right to left $($least significant digit to most significant$).$

For each character, determine its decimal value.

If the character is $\text{'}$A$\text{'}$ or $\text{'}$a$\text{'},$ its value is $10.$If the character is $\text{'}$B$\text{'}$ or $\text{'}$b$\text{'},$ its value is $11.$Otherwise, the value is the integer equivalent of the digit character.

Multiply this value by $12$ raised to the power of the position index $($starting from $0$ at the rightmost digit$)$ and add it to the resultant decimal number.

The final value of the resultant decimal number is the decimal equivalent of the duodecimal string.

Converting from Decimal to Duodecimal

To convert a number from decimal to duodecimal:

Steps:

Initialize an empty string for the duodecimal result.

While the decimal number is greater than zero:

Divide the decimal number by $12$ and store the remainder.Convert the remainder to the corresponding duodecimal digit:

If the remainder is $10,$ the digit is $\text{'}$A$\text{'}.$If the remainder is $11,$ the digit is $\text{'}$B$\text{'}.$Otherwise, the digit is the character representation of the remainder, e$.$g$.,$ the digit is $\text{'}9\text{'}$ when the remainder is $9.$

Insert the duodecimal digit at the beginning of the result string.Update the decimal number to be the quotient of the division by $12.$

If the initial decimal number was zero, the result is $"0".$

The final result string is the duodecimal representation of the decimal number.