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In C++, a `do-while loop` is a type of loop structure that ensures the code within its block executes at least once. This is because the condition check follows, rather than precedes, the block of code.

Here's how a `do-while loop` generally looks:

• Start with the `do` keyword.
• Place the code you want to execute inside curly braces `{}`.
• Follow the brace with the `while` keyword.
• The condition follows `while` in parentheses `()`.

This construct is particularly useful when a minimum of one iteration is necessary.
For instance, in our program, the block executes to print the initial value of `z`, regardless of its initial status and keeps updating until the loop condition fails.

It guarantees that `11` is output, then transitions into subsequent checks to determine whether additional iterations are warranted.

The modulus operator `%`, is a fundamental operation in C++ used to find the remainder of division between two numbers. In mathematical terms, ``` a % b ``` would give the remainder when `a` is divided by `b`.

This operator is quite handy for a multitude of applications, such as determining if one number is divisible by another. For example, if `a % b` evaluates to `0`, then `a` is divisible by `b`.

In our exercise, the modulus operator is part of the crucial exit condition for the `do-while loop`. Calculating ``` ((z - x) % 4) != 0 ``` is a check to decide whether another iteration will proceed. This involves checking if the difference between `z` and `x`, when divided by 4, doesn't leave a remainder.

It's a succinct way to iterate while ensuring particular mathematical conditions are met before exit.

Debugging in C++ is the analytical process of inspecting your code step by step to spot and rectify mistakes. It's vital for understanding how your code executes, which might not always be transparent while coding.

Engaging in step-by-step debugging can involve:

• Setting breakpoints strategically to halt execution.
• Inspecting variable values to ensure they are as expected at different points of execution.
• Stepping through code line by line to observe the flow and decision points.

This methodical approach can help reveal logical errors or confirm that your loops and conditions, like those in the do-while loop in our problem, perform as anticipated before arriving at erroneous outcomes.

Employing debug tools, particularly in complex programs, significantly eases this process by allowing a controlled execution and examination of the program mechanics. This ensures that your program functions smoothly and aligns with expected behaviors.