13. (15 points) Given a rectangular prism with the ratio of its length, width, and height as $4: 3: 2$, it is cut by a plane, and the cutting face is a hexagon (as shown in the figure). It is known that the minimum perimeter of all such hexagons is 36. Find the surface area of this rectangular prism.
【Analysis】According to the problem, the ratio of the length, width, and height of the rectangular prism is 4:3:2. When the perimeter of the hexagon is minimized, the six sides of the hexagon in the unfolded diagram should form a straight line. At this point, the length, width, and height of the rectangular prism can be calculated, and thus the surface area can also be determined.

【Solution】Solution: According to the analysis, the unfolded diagram of the rectangular prism is as follows: ( $AB$ and $CE$ are the same edge, $P$ and $Q$ are the same point)
So when the perimeter is minimized, the six sides of the hexagon in the unfolded diagram should form a straight line,
Therefore, the surface area of the rectangular prism is: $2 \times($ length $\times$ width + length $\times$ height + width $\times$ height $)=2 \times(2 \times 3+3 \times 4+4 \times 2) \times$ $\left[\left(\frac{36}{2+3+4}\right)^{2} \div 2\right]=416$,

Therefore, the answer is: 416.