
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 the sequence $\left\{a_{n}\right\}$ with the first term $a_{1}=1$, the sum of the first $n$ terms $S_{n}$ and the general term $a_{n}$ satisfying $a_{n}=\frac{2 S_{n}^{2}}{2 S_{n}-1}$ for $n \geq 2$, find:
(1) The general term formula for the sequence $\left\{S_{n}\right\}$.
(2) The general term formula for the sequence $\left\{a_{n}\right\}$.

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