【Analysis】The surface area is increased by two rectangles compared to the original cube.
As shown in the figure, $M J=12-4-3=5, M I=12$, by the Pythagorean theorem, $I J=13$ so the area of the new rectangle is $12 \times 13=156$
The sum of the two parts of the surface area is $6 \times 12^{2}+2 \times 156=1176$ square centimeters.
13. As shown in the figure, a cube with an edge length of 12 cm is cut with one slice. The cut is made along $I J$ and exits at $L K$, such that $A I=D L=4$ cm, $J F=K G=3$ cm, and the cross-section $I J K L$ is a rectangle. The cube is cut into two parts, and the sum of the surface areas of these two parts is $\qquad$ square centimeters.