
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 that the terms of a positive geometric sequence $\{a\_n\}$ have a sum of $S\_n$ for the first $n$ terms, with $S\_4=30$, and the slope of the line passing through points $P(n, \log\_2 a\_n)$ and $Q(n+2, \log\_2 a\_{n+1})$ $(n \in \mathbb{N}^*)$ is $1$. Let $b\_n = \frac{\log\_2 a\_{n+1}}{\log\_2^2 a\_{n+2} \cdot \log\_2^2 a\_n}$. Find the sum of the first $n$ terms of the sequence $\{b\_n\}$, denoted as $T\_n$.

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