
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 geometric sequence $\{a\_n\}$ with a common ratio not equal to $1$, its sum of the first $n$ terms is $S\_n$. It is known that $S\_6 = \frac{63}{32}$, and $-a\_2$, $a\_4$, $3a\_3$ form an arithmetic sequence.

1. Find the general term formula for the sequence $\{a\_n\}$.
2. Let sequence $\{b\_n\}$ satisfy $b\_n = n a\_n$, find the sum of the first $n$ terms, $T\_n$, for sequence $\{b\_n\}$.

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