
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 $a_{1}$ and $a_{5}$ are the two roots of the equation $x^{2}-12x+27=0$, and the sequence $\{a_{n}\}$ is an arithmetic sequence with a positive common difference. The sum of the first $n$ terms of the sequence $\{b_{n}\}$ is $T_{n}$, and $T_{n}=1-\frac{1}{2}b_{n}$ ($n\in\mathbb{N}^*$).

$(1)$ Find the general formula for the sequences $\{a_{n}\}$ and $\{b_{n}\}$.

$(2)$ Let $c_{n}=a_{n}b_{n}$, find the sum of the first $n$ terms of the sequence $\{c_{n}\}$, $S_{n}$.

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