
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 increasing geometric sequence $\{a_{n}\}$ with the sum of the first $n$ terms as $S_{n}$, and $S_{3}=13$, $a_{3}^{2}=3a_{4}$, an arithmetic sequence $\{b_{n}\}$ satisfies $b_{1}=a_{1}$, $b_{2}=a_{2}-1$.  
$(1)$ Find the general formulas for sequences $\{a_{n}\}$ and $\{b_{n}\}$;  
$(2)$ If ${c}_{n}=\left\{\begin{array}{l}-{a}_{n+1}{b}_{n},n\text{ is odd}\\{a}_{n}{b}_{n},n\text{ is even}\end{array}\right.$, determine the relationship between $c_{2n-1}+c_{2n}$ and $a_{2n}$, and find the sum of the first $20$ terms of sequence $\{c_{n}\}$.

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