
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 $\{a_{n}\}$ be an arithmetic sequence with a common difference of $d$, and $d=2a_{1}$, $a_{5}=9$.<br/>$(1)$ Find the general formula for the sequence $\{a_{n}\}$;<br/>$(2)$ Let $\{b_{n}\}$ be a sequence such that $a_{1}b_{1}+a_{2}b_{2}+\ldots +a_{n}b_{n}=3-\frac{2n+3}{{2}^{n}}$, find the sum of the first $n$ terms of $\{b_{n}\}$ denoted by $S_{n}$.

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