
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 the sum of its first $n$ terms denoted as $S_{n}$, and the sequence $\{b_{n}\}$ is a geometric sequence with a common ratio greater than $0$, and $a_{1}=b_{1}=2$, $S_{5}=5b_{2}=a_{11}+b_{3}$.  
$(1)$ Find the general formulas for $\{a_{n}\}$ and $\{b_{n}\}$;  
$(2)$ Let ${c}_{n}=\frac{{a}_{n}}{{b}_{n}}(n∈{N}^{*})$, determine 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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