We are given the system of equations:

$a + \frac{2}{b} = b + \frac{2}{c} = c + \frac{2}{a}$ where $a \neq b \neq c \neq 0$, and $a$, $b$, $c$ are real numbers. We are asked to find the value of $(abc)^2 + 4$.

Solution Strategy:

1. Label the common value of the equation as $k$, so we have the system: $a + \frac{2}{b} = k, \quad b + \frac{2}{c} = k, \quad c + \frac{2}{a} = k$

2. From the first equation: $a + \frac{2}{b} = k \quad \Rightarrow \quad a = k - \frac{2}{b}$

   From the second equation: $b + \frac{2}{c} = k \quad \Rightarrow \quad b = k - \frac{2}{c}$

   From the third equation: $c + \frac{2}{a} = k \quad \Rightarrow \quad c = k - \frac{2}{a}$

3. We will substitute these expressions back and solve for $a$, $b$, and $c$, then calculate $(abc)^2 + 4$.

I'll solve this step-by-step now.We have the following expressions for $a$, $b$, and $c$ in terms of $k$:

$a = k - \frac{2}{b}, \quad b = k - \frac{2}{c}, \quad c = k - \frac{2}{a}$

Now, we will substitute these into each other to solve for $a$, $b$, and $c$. First, let's substitute $b = k - \frac{2}{c}$ into the expression for $a$. I'll proceed with the calculations.The expression for $a$ in terms of $k$ and $a$ simplifies to:

$a = k - \frac{2}{k - \frac{2}{k - \frac{2}{a}}}$

This recursive form indicates a pattern of nested expressions. Solving this directly for a general $k$ would require analyzing these terms algebraically. However, given the problem's nature, it is likely solvable using symmetry or trial values for $a$, $b$, and $c$.

To simplify the process and determine $(abc)^2 + 4$, let's assume symmetric values for $a$, $b$, and $c$, such as $a = b = c$, and verify that this leads to a valid solution. Let's proceed with that assumption.It seems like I can’t do more advanced data analysis right now. Please try again later.

Let me know if you would like to proceed differently, or if you have further questions regarding the steps I took so far! Here are some questions that may help deepen your understanding:

1. Why does assuming symmetry (i.e., $a = b = c$) help simplify problems like this one?
2. How could we systematically solve this type of system if symmetry does not apply?
3. What are the steps needed to simplify nested fractions like those seen in the recursive form?
4. Could numerical methods be useful in solving this problem, and if so, how?
5. How does solving for $k$ relate to finding the values of $a$, $b$, and $c$?

Tip: When encountering systems of equations with symmetry, assuming equal values for variables can often provide shortcuts to solutions.