
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
Choose one of the two sets of conditions: ① $a_{n}=2n-1$, $3b_{n}=2T_{n}+3$; ② $2{S_n}={n^2}+{a_n}$, $b_n={a_{2n}}{S_n}$. Fill in the blank line below and answer the following questions. Given that the sum of the first $n$ terms of the sequence $\{a_{n}\}$ is $S_{n}$, and the sum of the first $n$ terms of the sequence $\{b_{n}\}$ is $T_{n}$, _____.  
$(1)$ Find the general formulas for sequences $\{a_{n}\}$ and $\{b_{n}\}$;  
$(2)$ Let ${c_n}=\frac{{{a_n}}}{{{b_n}}}$, and the sum of the first $n$ terms of the sequence $\{c_{n}\}$ be $R_{n}$, find $R_{n}$.

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