
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
For the sequence $\{a_n\}$, ($n \in \mathbb{N}^+$, $a_n \in \mathbb{N}^+$), if $b_k$ is the maximum value among $a_1$, $a_2$, ..., $a_k$ ($k=1, 2, ..., n$), then the sequence $\{b_n\}$ is called the "convex sequence" of $\{a_n\}$. For example, the "convex sequence" of the sequence 2, 1, 3, 7, 5 is 2, 2, 3, 7, 7. Based on this definition, the correct statements are:  
① The "convex sequence" of a decreasing sequence $\{a_n\}$ is a constant sequence;  
② There does not exist a sequence $\{a_n\}$ whose "convex sequence" is the sequence $\{a_n\}$ itself;  
③ The "convex sequence" of any sequence $\{a_n\}$ is an increasing sequence;  
④ The number of all sequences $\{a_n\}$ whose "convex sequence" is 1, 3, 3, 9, is 3.

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