
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 sequence $\{a\_n\}$ with its first $n$ terms sum $S\_n$, where $a\_1=1$, and $S_{n+1} = 4a\_n + 2 (n \in \mathbb{N}^*)$.

1. If $b\_n = a_{n+1} - 2a\_n$, find $b\_n$.
2. If $c\_n = \frac{1}{a_{n+1} - 2a\_n}$, find the sum of the first 6 terms of $\{c\_n\}$, denoted as $T\_6$.
3. If $d\_n = \frac{a\_n}{2^n}$, find the general term of the sequence $\{d\_n\}$.

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