
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 $\{a_{n}\}$ is an arithmetic progression, $\{b_{n}\}$ is a geometric progression, and $b_{2}=3$, $b_{3}=9$, $a_{1}=b_{1}$, $a_{14}=b_{4}$.  
$(1)$ Find the general formula for $\{a_{n}\}$;  
$(2)$ Let $c_{n}=a_{n}+(-1)^{n}b_{n}$, find the sum of the first $2n$ terms of the sequence $\{c_{n}\}$.

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