
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 sequence $\{a_{n}\}$ is an arithmetic sequence with a positive common difference, and the sum of its first $n$ terms is ${S}_{n}={n}^{2}+n$; the sequence $\{b_{n}\}$, where all terms are positive, is a geometric sequence satisfying ${b}_{1}{b}_{3}=\frac{1}{16}$ and ${b}_{5}=\frac{1}{32}$.  
$(1)$ Find the general formulas for sequences $\{a_{n}\}$ and $\{b_{n}\}$;  
$(2)$ Let $c_{n}=2a_{n}-b_{n}$, find the sum of the first $n$ terms of the sequence $\{c_{n}\}$ denoted by $T_{n}$.

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