
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
The sequence $\{a\_n\}$ is an arithmetic sequence with a positive common difference. $a\_2$ and $a\_5$ are the two real roots of the equation $x^{2}-12x+27=0$. The sequence $\{b\_n\}$ satisfies $3^{n-1}b\_n=na_{n+1}-(n-1)a\_n$.

1. Find $a\_n$ and $b\_n$;
2. Let $T\_n$ be the sum of the first $n$ terms of the sequence $\{b\_n\}$. Find $T\_n$ and the maximum value of $n$ such that $T\_n < 7$.

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