
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 integer sequence \(\left\{a_{n}\right\}\) satisfying the condition: For any positive integer \(n\), let's select several numbers from the set \(\left\{a_{1}, a_{2}, \cdots, a_{n}\right\}\) without repetition. The absolute values of the results obtained by adding and subtracting these numbers are mutually different positive integers. These positive integers and the sequence \(\{a_1, a_2, \cdots, a_n\}\) together exactly form the set of all positive integers from 1 to \(S_{n}\), where \(S_{n}\) is the sum of the first \(n\) terms of the sequence \(\left\{a_{n}\right\}\).

(1) Find the values of \(a_{1}\) and \(a_{2}\);

(2) Find the general term formula for the sequence \(\left\{a_{n}\right\}\).

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