Example 7 (2006 Beijing College Entrance Examination Final Question) In the sequence $\left\{a_{n}\right\}$, if $a_{1}, a_{2}$ are positive integers, and $a_{n}=\left|a_{n-1}-a_{n-2}\right|$, $n=3,4,5, \cdots$, then $\left\{a_{n}\right\}$ is called an "absolute difference sequence".
(1) Give an example of an "absolute difference sequence" whose first five terms are not zero (only the first ten terms are required to be written);
(2) If in the "absolute difference sequence" $\left\{a_{n}\right\}$, $a_{20}=3, a_{21}=$ 0, and the sequence $\left\{b_{n}\right\}$ satisfies $b_{n}=a_{n}+a_{n+1}+a_{n+2}, n=1,2$, $3, \cdots$, determine whether the limits of $a_{n}$ and $b_{n}$ exist as $n \rightarrow \infty$. If they exist, find their limit values;
(3) Prove: Any "absolute difference sequence" contains infinitely many zero terms.
(1) $a_{1}=3, a_{2}=1, a_{3}=2, a_{4}=1, a_{5}=1$, $a_{6}=0, a_{7}=1, a_{8}=1, a_{9}=0, a_{10}=1$. (The answer is not unique.)
(2) Since in the absolute difference sequence $\left\{a_{n}\right\}$, $a_{20}=3, a_{21}=$ 0, starting from the 20th term, the sequence is $a_{20}=3, a_{21}=$ $0, a_{22}=3, a_{23}=3, a_{24}=0, a_{25}=3, a_{26}=3, a_{27}=$ $0, \cdots$.

That is, starting from the 20th term, every three adjacent terms cyclically take the values $3,0,3$, so when $n \rightarrow \infty$, the limit of $a_{n}$ does not exist.

When $n \geqslant 20$, $b_{n}=a_{n}+a_{n+1}+a_{n+2}=6$, so $\lim _{n \rightarrow \infty} b_{n}=6$.
(3) Proof: According to the definition, the sequence $\left\{a_{n}\right\}$ must have a zero term after a finite number of terms, as proven below:

Assume that $\left\{a_{n}\right\}$ has no zero terms. Since $a_{n}=\mid a_{n-1}-$ $a_{n-2} \mid$, for any $n$, we have $a_{n} \geqslant 1$, thus

When $a_{n-1} \geqslant a_{n-2}$, $a_{n}=a_{n-1}-a_{n-2} \leqslant a_{n-1}-1$ $(n \geqslant 3)$;

When $a_{n-1}a_{2 n}\right)$, \\
$a_{2 n}\left(a_{2 n-1}0(n=1,2,3, \cdots)$, which contradicts the assumption, thus $\left\{a_{n}\right\}$ must have a zero term.

If the first zero term appears at the $n$-th term, i.e., $a_{n}=0$, let $a_{n-1}=A(A \neq 0)$, then $a_{n+1}=A, a_{n+2}=0, \cdots$, hence starting from the $n$-th term, every three adjacent terms cyclically take the values 0, and the absolute difference sequence $\left|a_{n}\right|$ has infinitely many zero terms.