
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 a sequence $\{a\_n\}$ with the sum of its first $n$ terms denoted as $S\_n$. It is known that $a\_1=1$ and $a_{n+1}= \frac {1}{n}S\_n+ \frac {1}{2}(n+1)\ (n\in\mathbb{N}^*)$.

1. Find the general formula for the sequence $\{a\_n\}$.
2. Let $a\_n=2^{n-1}b\_n\ (n\in\mathbb{N}^*)$, and denote the sum of the first $n$ terms of the sequence $\{b\_n\}$ as $T\_n$. If $T\_n\geqslant k- \frac {9}{2^{n}}$ holds for all $n\in\mathbb{N}^*$, find the maximum value of the integer $k$.

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