
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
The sequence $\{a_n\}$ is a geometric sequence with positive terms, $a_1=2$, and $\frac{3}{8}a_4$ is the arithmetic mean of $a_2$ and $a_3$. The sum of the first $n$ terms of the sequence $\{b_n\}$ is denoted as $S_n$, and $2b_2=b_1+b_3$. $\sqrt{S_n}$ forms an arithmetic sequence with a common difference of $1$.
1. Find the sum of the first $n$ terms, $T_n$, of the sequence $\{na_n\}$.
2. Find the general formula for the $n$-th term of the sequence $\{b_n\}$.
3. Determine whether there exists $n \in \mathbb{N}^*$ such that $S_n=a_n$ holds. If it exists, find all such $n$; if not, explain why.

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