
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 $\{a_{n}\}$ is an arithmetic sequence, $\{b_{n}\}$ is a geometric sequence, the sum of the first $n$ terms of $\{b_{n}\}$ is ${S_n}=3•{2^n}-3$, $a_{1}=b_{1}$, and $a_{7}+a_{16}=b_{5}$.  
$(1)$ Find the general formulas for sequences $\{a_{n}\}$ and $\{b_{n}\}$.  
$(2)$ Let ${c_n}=\frac{{{a_n}-1}}{{{b_n}}}$. 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}
user