
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 an arithmetic sequence $\{a_{n}\}$ with a common difference $d$ that is not equal to $0$, the sum of the first $n$ terms is $S_{n}$, $a_{3}=6$, and $\frac{S_{5}}{S_{9}}=\frac{1}{3}$. Find:<br/>
$(1)$ The general formula for the sequence $\{a_{n}\}$;<br/>
$(2)$ If the sequence ${b_n}={2^{{a_n}}}$ and $c_{n}=a_{n}+b_{n}$, find 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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