
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Given two sequences $\{a_n\}$ and $\{b_n\}$ are both monotonically increasing. If these two sequences are combined and arranged in ascending order (identical items are considered as one item), a new sequence $\{c_n\}$ is obtained.  
(1) Suppose $\{a_n\}$ and $\{b_n\}$ are arithmetic and geometric sequences, respectively, with $a_1=b_1=1$, $a_2=b_3$, and $a_6=b_5$. Find $c_{20}$.  
(2) Suppose the first term of $\{a_n\}$ is 1, and each term is a positive integer, with $b_n=3^n$. If the new sequence $\{c_n\}$ is an arithmetic sequence, find the sum of the first $n$ terms of $\{c_n\}$, $S_n$.  
(3) Let $b_n=q^{n-1}$ (where $q$ is a positive integer not less than 2) and $c_1=b_1$. Is there an arithmetic sequence $\{a_n\}$ such that for any $n\in\mathbb{N}^*$, the number of terms of $\{a_n\}$ between $b_n$ and $b_{n+1}$ is always $b_n$? If it exists, please provide an arithmetic sequence $\{a_n\}$ that satisfies the conditions; if not, please explain why.

Present the answer in LaTex format: \boxed{Your answer}
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