
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 an geometric sequence $\{a\_n\}$ with the sum of the first $n$ terms denoted as $S\_n$. It is known that $S\_2 = \frac{3}{2}a\_2 - 1$ and $S\_3 = \frac{3}{2}a\_3 - 1$.

1. Find the general formula for the $n$-th term of the sequence.
2. Insert $n$ numbers between $a\_n$ and $a\_{n+1}$ to form an arithmetic sequence with a common difference of $d\_n$. Denote $T\_n$ as the sum of the first $n$ terms of the sequence $\{ \frac{1}{d\_n} \}$. Find the maximum positive integer value of $n$ that satisfies $\frac{8}{5}T\_n + \frac{n}{5 \times 3^{n-1}} \leqslant \frac{40}{27}$.

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