
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 $\{a_n\}$, where $a_1=2$, $a_2=4$, and $S_n$ represents the sum of the first $n$ terms of the sequence. For any $n > 1$, $n \in \mathbb{N}^*$, $S_{n+1} + S_{n-1} = 2(S_n + 1)$.

1. Find the general term formula for the sequence $\{a_n\}$.
2. Let $b_n = \frac{n}{2^{a_n}}$, find the sum of the first $n$ terms, denoted as $T_n$, for the sequence $\{b_n\}$.

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