
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 $\{a_{n}\}$ and a geometric sequence $\{b_{n}\}$ satisfying $a_{1}=2$, $b_{1}=1$, $a_{2}+a_{3}=10$, $b_{2}b_{3}=-a_{4}$. Find:<br/>
$(1)$ The general formulas for sequences $\{a_{n}\}$ and $\{b_{n}\}$;<br/>
$(2)$ Let the sequence $\{c_{n}\}$ satisfy $c_{n}=a_{n}+b_{n}$, find the sum $c_{1}+c_{3}+c_{5}+\ldots +c_{2n-1}$.

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