A laser is placed at the point $(2,4)$. The beam travels in a straight line, hits and bounces off the $y$-axis, then hits and bounces off the $x$-axis, then hits the point $(4,4)$. What is the total distance the beam will travel along this path?

1. Reflecting the target point $(4,4)$ over the $x$-axis gives $(4,-4)$, then reflecting over the $y$-axis gives $(-4,-4)$.
2. The total distance equals the straight-line distance from $(2,4)$ to $(-4,-4)$: $\sqrt{6^2+8^2}=\sqrt{100}=10$.
3. The answer is $\boxed{10}$.