
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 function $f(x)=x^{2}+(a-1)x+b+1$, when $x\in[b,a]$, the graph of the function $f(x)$ is symmetric about the $y$-axis. The sum of the first $n$ terms of the sequence $\{a_{n}\}$ is denoted as $S_{n}$, and $S_{n}=f(n+1)-1$.

1. Find the general formula for the sequence $\{a_{n}\}$.
2. Let $b_{n}= \frac{a_{n}}{2^{n}}$, find the sum of the first $n$ terms of the sequence $\{b_{n}\}$, denoted as $T_{n}$.

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