
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 an infinite sequence $\{a_n\}$, where each term is a positive integer, and $S_n$ is the sum of the first $n$ terms of the sequence $\{a_n\}$.
1. If the sequence $\{a_n\}$ is an arithmetic sequence, and for any positive integer $n$, $S_n^3 = (S_n)^3$ holds, find the general formula for the sequence $\{a_n\}$.
2. For any positive integer $n$, by selecting several distinct numbers from the set $\{a_1, a_2, \ldots, a_n\}$ and performing addition and subtraction operations among these numbers, the absolute values of the resulting numbers are distinct positive integers. Moreover, these positive integers, together with $a_1, a_2, \ldots, a_n$, exactly constitute the set of all positive integers from 1 to $S_n$.
   (i) Find the values of $a_1, a_2$;
   (ii) Find the general formula for the sequence $\{a_n\}$.

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