A rectangular wooden block has a square top and bottom, its volume is $576$, and the surface area of its vertical sides is $384$. Find the sum of the lengths of all twelve of the edges of the block.
1. Let the side length of the square base be \( s \) and the height of the block be \( h \). We are given two pieces of information:
   \[
   s^2 \cdot h = 576
   \]
   \[
   4sh = 384
   \]

2. From the second equation, solve for \( sh \):
   \[
   4sh = 384 \implies sh = \frac{384}{4} = 96
   \]

3. Substitute \( sh = 96 \) into the first equation:
   \[
   s^2 \cdot h = 576
   \]
   \[
   s \cdot (sh) = 576 \implies s \cdot 96 = 576
   \]
   \[
   s = \frac{576}{96} = 6
   \]

4. Now, substitute \( s = 6 \) back into \( sh = 96 \) to find \( h \):
   \[
   6h = 96 \implies h = \frac{96}{6} = 16
   \]

5. The block has 12 edges: 4 edges of length \( s \) on the top, 4 edges of length \( s \) on the bottom, and 4 vertical edges of length \( h \). Therefore, the sum of the lengths of all twelve edges is:
   \[
   4s + 4s + 4h = 8s + 4h
   \]
   Substitute \( s = 6 \) and \( h = 16 \):
   \[
   8 \cdot 6 + 4 \cdot 16 = 48 + 64 = 112
   \]

The final answer is \(\boxed{112}\).