7. Let the four vertices of a regular tetrahedron be $A, B, C, D$, with each edge length being 1 meter. A small insect starts from point $A$ and moves according to the following rule: at each vertex, it chooses one of the three edges connected to that vertex with equal probability, and crawls all the way to the end of that edge. What is the probability that after crawling 7 meters, it is exactly at vertex $A$? $\qquad$
7. Solution: Let $a_{n}$ denote the probability that the bug returns to point $A$ after walking $n$ meters $(n=0,1,2 \cdots \cdots)$. If the bug has walked $n-1$ meters and is not at $A$, the probability is $1-a_{n-1}$. The probability of moving from a point outside $A$ to $A$ is $\frac{1}{3}$. Therefore, we have $a_{n}=\frac{1}{3}\left(1-a_{n-1}\right)$. Given that $a_{0}=1$, we can calculate that $a_{7}=\frac{182}{729}$. Thus, the required probability is $\frac{182}{729}$.