
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 its first $n$ terms sum $S\_n$, where $a\_1=1$, and $2nS_{n+1}-2(n+1)S_n=n(n+1) (n\in \mathbb{N}^*)$. Sequence $\{b\_n\}$ satisfied $b_{n+2}-2b_{n+1}+b_n=0 (n\in \mathbb{N}^*)$, with $b_3=5$, and the sum of its first 9 terms is 63.

1. Find the general term formulas for both sequences $\{a\_n\}$ and $\{b\_n\}$.
2. Let $c\_n=\frac{b\_n}{a\_n}+\frac{a\_n}{b\_n}$. If the sum of the first $n$ terms of sequence $\{c\_n\}$, denoted as $T\_n$, satisfies $T\_n-2n \in [a,b]$ for any positive integer $n$, find the minimum value of $b-a$.

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