
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
(1) Let the sequence $\{a_n\}$ be an arithmetic sequence with first term $a_1$ ($a_1 > 0$) and common difference $4$. Denote the sum of the first $n$ terms as $S_n$. It is given that $\sqrt{S_1}$, $\sqrt{S_2}$, and $\sqrt{S_3}$ form an arithmetic sequence. Find the general formula for the $n$-th term of the sequence $\{a_n\}$;

(2) Let the sequence $\{a_n\}$ have positive terms, and denote the sum of the first $n$ terms as $S_n$. It is given that $S_n - S_{n-1} = \sqrt{S_n} + \sqrt{S_{n-1}}$ for $n \ge 2$, and $a_1 = 1$. Find the general formula for the $n$-th term of the sequence $\{a_n\}$.

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