Example 3 (1) Given the geometric sequences $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$ satisfy:
$$
\begin{array}{l}
a_{1}=a>0, b_{1}-a_{1}=1, \\
b_{2}-a_{2}=2, b_{3}-a_{3}=3 .
\end{array}
$$

If the sequence $\left\{a_{n}\right\}$ exists and is unique, find the value of $a$.
(2) Do there exist two geometric sequences $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$ such that $b_{1}-a_{1}, b_{2}-a_{2}, b_{3}-a_{3}, b_{4}-a_{4}$ form an arithmetic sequence with a non-zero common difference? If they exist, find the general terms of $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$; if not, explain the reason.
【Analysis】"The common ratio of a geometric sequence cannot be 0" is a common knowledge. If this is asked directly, it would become a "junk question". However, if this concept is appropriately hidden in the problem-solving process, it could become a good question.

This problem is from the 2011 Jiangxi Province College Entrance Examination Mathematics Test, which hides the concept that "the common ratio of a geometric sequence cannot be 0" in the problem statement. According to the statistics from the grading group, the vast majority of candidates attributed the "uniqueness" of the problem to whether the discriminant is zero, which did not succeed, resulting in a very low score rate for the problem.
Solution (1) Let the common ratio of $\left\{a_{n}\right\}$ be $q$. Then
$$
b_{1}=1+a, b_{2}=2+a q, b_{3}=3+a q^{2} \text {. }
$$

Since $b_{1} 、 b_{2} 、 b_{3}$ form a geometric sequence, we have
$$
\begin{array}{l}
(2+a q)^{2}=(1+a)\left(3+a q^{2}\right) \\
\Rightarrow a q^{2}-4 a q+3 a-1=0 .
\end{array}
$$

Since $a>0 \Rightarrow \Delta=4 a^{2}+4 a>0$
$\Rightarrow$ the equation has two distinct real roots.
Furthermore, since $\left\{a_{n}\right\}$ exists and is unique, the equation must have one root as 0. Substituting $q=0$ into the equation gives $a=\frac{1}{3}$.
(2) Assume there exist two geometric sequences $\left\{a_{n}\right\} 、\left\{b_{n}\right\}$ that satisfy the conditions, with the common ratio of $\left\{a_{n}\right\}$ being $q_{1}$ and the common ratio of $\left\{b_{n}\right\}$ being $q_{2}$. Then
$$
\begin{array}{l}
b_{2}-a_{2}=b_{1} q_{2}-a_{1} q_{1}, \\
b_{3}-a_{3}=b_{1} q_{2}^{2}-a_{1} q_{1}^{2}, \\
b_{4}-a_{4}=b_{1} q_{2}^{3}-a_{1} q_{1}^{3} .
\end{array}
$$

By the problem's conditions, we have
$$
\begin{array}{l} 
2\left(b_{1} q_{2}-a_{1} q_{1}\right) \\
=b_{1}-a_{1}+\left(b_{1} q_{2}^{2}-a_{1} q_{1}^{2}\right), \\
2\left(b_{1} q_{2}^{2}-a_{1} q_{1}^{2}\right) \\
=b_{1} q_{2}-a_{1} q_{1}+\left(b_{1} q_{2}^{3}-a_{1} q_{1}^{3}\right), \\
\text { i.e., }\left\{\begin{array}{l}
b_{1}\left(q_{2}-1\right)^{2}-a_{1}\left(q_{1}-1\right)^{2}=0, \\
b_{1} q_{2}\left(q_{2}-1\right)^{2}-a_{1} q_{1}\left(q_{1}-1\right)^{2}=0 .
\end{array}\right.
\end{array}
$$
(1) $\times q_{2}$-(2) gives
$$
a_{1}\left(q_{1}-q_{2}\right)\left(q_{1}-1\right)^{2}=0 \text {. }
$$

Since $a_{1} \neq 0$, we have $q_{1}=q_{2}$ or $q_{1}=1$.
When $q_{1}=q_{2}$, from equations (1) and (2) we get
$$
b_{1}=a_{1} \text { or } q_{1}=q_{2}=1 \text {, }
$$

In this case, $\left(b_{2}-a_{2}\right)-\left(b_{1}-a_{1}\right)=0$ contradicts the condition that the common difference is not 0.

When $q_{1}=1$, from equations (1) and (2) we get $b_{1}=0$ or $q_{2}=$ 1, in which case $\left(b_{2}-a_{2}\right)-\left(b_{1}-a_{1}\right)=0$ also contradicts the condition that the common difference is not 0.

Therefore, there do not exist two geometric sequences $\left\{a_{n}\right\} 、\left\{b_{n}\right\}$ that satisfy the conditions.

In "Dream of the Red Chamber," it is said that the magic jade is "coming without a trace, going without a trace, leaning on an ancient pine under the Green Peak; to seek it, mountains are many, but a smile greets us at the door," describing such a realm. And the problem setters often need to discover and find such a "magic jade" in the structure of mathematics.