
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 arithmetic sequence $\{a\_n\}$ has a sum of the first $n$ terms $S\_n$, and $2^{n+1}$, $S\_n$, $a$ form an arithmetic sequence ($n \in \mathbb{N}^*$).

(1) Find the value of $a$ and the general term formula of the sequence $\{a\_n\}$.

(2) If $b\_n = (1 - an) \log\_2(a\_n a\_{n+1})$, find the sum of the first $n$ terms of the sequence $\{\frac{1}{b\_n}\}$, denoted as $K\_n$.

(3) If $c\_n = -(an + 1) a\_n$, find the sum of the first $n$ terms of the sequence $\{c\_n\}$, denoted as $T\_n$.

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