
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 a sequence $\{a_n\}$, where $a_1=1$, insert 1 number between $a_1$ and $a_2$, 2 numbers between $a_2$ and $a_3$, 3 numbers between $a_3$ and $a_4$, ..., $n$ numbers between $a_n$ and $a_{n+1}$. All the inserted numbers and the original sequence $\{a_n\}$ form a new positive arithmetic sequence $\{b_n\}$.

(1) If $a_4=19$, find the general formula for $\{b_n\}$;

(2) Let $S_n$ denote the sum of the first $n$ terms of $\{b_n\}$, and suppose $\sqrt{2S_n+\lambda}=b_n+\mu$ ($\lambda, \mu$ are constants), find the general formula for $\{a_n\}$.

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