We are given the system of equations:

$a - 7b + 8c = 4 \quad \text{(1)}$ $8a + 4b - c = 7 \quad \text{(2)}$

We need to find the value of $a^2 + b^2 + c^2$.

Step 1: Solve the system of equations

First, let's solve for $a$, $b$, and $c$.

Equation (1): $a - 7b + 8c = 4 \quad \text{(1)}$

Equation (2): $8a + 4b - c = 7 \quad \text{(2)}$

We can solve this system using the substitution or elimination method.

Substitution/Elimination Method:

From equation (2), solve for $c$:

$c = 8a + 4b - 7$

Now substitute this expression for $c$ into equation (1):

$a - 7b + 8(8a + 4b - 7) = 4$ Simplify the terms:

$a - 7b + 64a + 32b - 56 = 4$ $65a + 25b - 56 = 4$ $65a + 25b = 60$

Now, divide the entire equation by 5:

$13a + 5b = 12 \quad \text{(3)}$

Step 2: Solve for $b$

Solve equation (3) for $b$:

$5b = 12 - 13a$ $b = \frac{12 - 13a}{5}$

Step 3: Substitute into the expression for $c$

Now substitute $b = \frac{12 - 13a}{5}$ into the expression for $c$:

$c = 8a + 4\left(\frac{12 - 13a}{5}\right) - 7$ Simplify:

$c = 8a + \frac{48 - 52a}{5} - 7$ $c = 8a + \frac{48 - 52a - 35}{5}$ $c = 8a + \frac{13 - 52a}{5}$ $c = \frac{40a + 13 - 52a}{5}$ $c = \frac{-12a + 13}{5}$

Step 4: Calculate $a^2 + b^2 + c^2$

Now we have expressions for $b$ and $c$ in terms of $a$:

$b = \frac{12 - 13a}{5}, \quad c = \frac{-12a + 13}{5}$

To find $a^2 + b^2 + c^2$, we'll use these expressions:

$a^2 + b^2 + c^2 = a^2 + \left( \frac{12 - 13a}{5} \right)^2 + \left( \frac{-12a + 13}{5} \right)^2$

This expression can be simplified, but since it's a bit involved, let's check if there’s a more efficient approach to finalize the value of $a^2 + b^2 + c^2$.