
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
Let $m > 3$, for a finite sequence $\{a_n\}$ ($n=1,2,\ldots,m$), let $b_k$ be the maximum value among $a_1, a_2, \ldots, a_k$, and call the sequence $\{b_n\}$ the "innovation sequence" of $\{a_n\}$. The number of distinct terms in the sequence $\{b_n\}$ is called the "innovation order" of $\{a_n\}$. For example, the innovation sequence of the sequence 2, 1, 3, 7, 5 is 2, 2, 3, 7, 7, and its innovation order is 3. Consider all permutations of the natural numbers 1, 2, ...m ($m > 3$), each permutation is considered as a finite sequence $\{C_n\}$.
(1) If $m=5$, list all sequences $\{C_n\}$ whose innovation sequence is 3, 4, 4, 5, 5;
(2) Is there a sequence $\{C_n\}$ whose innovation sequence is an arithmetic sequence? If so, find all such sequences $\{C_n\}$; if not, explain why.

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