
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 $\{a_{n}\}$ that satisfies $a_{3}=2$, and the sum of its first three terms $S_{3}=\frac{9}{2}$.
(1) Find the general term formula for $\{a_{n}\}$;
(2) Suppose a geometric sequence $\{b_{n}\}$ satisfies $b_{1}=a_{1}$, $b_{4}=a_{15}$, and let $c_{n}=a_{n}+b_{n}$, find the sum $T_{n}$ of the first $n$ terms of $\{c_{n}\}$.

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