
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 its first $n$ terms as $S_{n}$, and $2^{n+1}$, $S_{n}$, $a$ form an arithmetic sequence ($n∈\mathbb{N}^∗$).
(1) Find the value of $a$ and the general formula for 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 $T_{n}$ for the sequence ${\frac{1}{b_{n}}}$.

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