
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 increasing sequence $\{a\_n\}$, with $a\_1=2$, the sum of its first $n$ terms is $S\_n$, and it satisfies the equation $a\_n^2+2=3(S\_n+S_{n-1}) (n\geqslant 2)$.

1. Find the value of $a\_2$;
2. Find the general term formula for the sequence $\{a\_n\}$;
3. If sequence $\{b\_n\}$ satisfies $\log\_2 \frac{b\_n}{a\_n}=n$, find its sum of the first $n$ terms $T\_n$.

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