
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 \(\left\{a_{n}\right\}\) satisfying that \(a_{1}\), \(a_{2}+1\), and \(a_{3}\) form an arithmetic progression, and for any positive integer \(n\), 
\[ 
S_{n} = \frac{1}{2}a_{n+1} - 2^{n} + \frac{3}{2},
\]
where \(S_{n}\) is the sum of the first \(n\) terms of the sequence \(\left\{a_{n}\right\}\). Find:
(1) The values of \(a_{1}\), \(a_{2}\), and \(a_{3}\);
(2) The general term formula of the sequence \(\left\{a_{n}\right\}\).

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