
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
Let $S_{n}$ be the sum of the first $n$ terms of the positive sequence ${a_{n}}$, and $S_{n}= \frac{1}{4}a_{n}^{2}+ \frac{1}{2}a_{n}- \frac{3}{4}$.
(1) Find the value of $a_{1}$;
(2) Find the general term formula for the sequence ${a_{n}}$;
(3) Given $b_{n}=2^{n}$, find the value of $T_{n}=a_{1}b_{1}+a_{2}b_{2}+\ldots+a_{n}b_{n}$.

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