13. (15 points) As shown in the figure, the area of hexagon $A B C D E F$ is 2010 square centimeters. It is known that the areas of $\triangle A B C, \triangle B C D, \triangle C D E, \triangle D E F, \triangle E F A, \triangle F A B$ are all 335 square centimeters, and the sum of the areas of the 6 shaded triangles is 670 square centimeters. Find the area of hexagon $A_{1} B_{1} C_{1} D_{1} E_{1} F_{1}$.
【Analysis】The area of hexagon $A_{1} B_{1} C_{1} D_{1} E_{1} F_{1}$ = the area of hexagon $A B C D E F$ - the area of the region between the two hexagons, from which we can solve the problem.

【Solution】According to the principle of inclusion-exclusion:
The area of the region between the two hexagons $=$
$$
\begin{aligned}
& (335 \times 6+670) \div 2 \\
= & 2680 \div 2 \\
= & 1340
\end{aligned}
$$

Therefore: The area of hexagon $A_{1} B_{1} C_{1} D_{1} E_{1} F_{1}$ $=2010-1340=670$
Answer: The area of hexagon $A_{1} B_{1} C_{1} D_{1} E_{1} F_{1}$ is 670.