In the coordinate plane, let $A=(0,0)$, $B=(20,0)$, $C=(20,20)$, and $D=(0,20)$. When a beam hits a side of this square, we can imagine "reflecting" the square over that side and allowing the beam to continue in a straight line through the reflected square. Repeating this reflection process three times, the beam reaches the point $M=(50,40)$ in the expanded plane of reflected squares. Since the beam’s path is effectively a straight line from the starting point to $M$, we can calculate the total distance traveled as the direct distance between the points $(0,0)$ and $(50,40)$, which is: $$\sqrt{(50-0)^2+(40-0)^2}=10\sqrt{5^2+4^2}=10\sqrt{41}$$Thus, the answer is $10\sqrt{41}$. [asy][asy] // Asymptote code to label points, draw lines, and include axes with modifications size(300); // Set canvas size // Define the points pair A = (0,0); pair B = (20,0); pair C = (20,20); pair D = (0,20); pair Ap = (40,0); pair Dp = (40,20); pair Bp = (20,40); pair App = (40,40); pair Bpp = (60,40); pair Cp = (60,20); pair M = (50,40); // Draw the points and label them dot(A); label("$A$", A, SW); dot(B); label("$B$", B, SE); dot(C); label("$C$", C, NE); dot(D); label("$D$", D, NW); dot(Ap); label("$A'$", Ap, S); dot(Dp); label("$D'$", Dp, NE); dot(Bp); label("$B'$", Bp, NE); dot(App); label("$A''$", App, NE); dot(Bpp); label("$B''$", Bpp, NE); dot(Cp); label("$C'$", Cp, NE); dot(M); label("$M$", M, NE); // Draw the specified lines draw(A--M, red+linewidth(1.5)); // Line AM in red and thicker draw(A--D, blue); // Line AD draw(B--Bp, blue); // Line BB' draw(Ap--App, blue); // Line A'A'' draw(Cp--Bpp, blue); // Line C'B'' draw(D--Cp, blue); // Line DC' draw(Bp--Bpp, blue); // Line B'B'' // Include the axes without maximum values xaxis("", Bottom, NoTicks); yaxis("", Left, NoTicks); [/asy][/asy]

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$ABCD$ is a square with side length 20. A light beam is radiated from $A$ and intersects sides $BC,CD,DA$ respectively and reaches the midpoint of side $AB$. What is the length of the path that the beam has taken? Proposed by Mahdi Etesamifard - Iran