
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
Let $S_{n}$ be the sum of the first $n$ terms of the sequence $\{a_{n}\}$, $a_{2}=5$, $S_{n+1}=S_{n}+a_{n}+4$; $\{b_{n}\}$ is a geometric sequence, $b_{2}=9$, $b_{1}+b_{3}=30$, with a common ratio $q \gt 1$.  
$(1)$ Find the general formulas for sequences $\{a_{n}\}$ and $\{b_{n}\}$;  
$(2)$ Let all terms of sequences $\{a_{n}\}$ and $\{b_{n}\}$ form sets $A$ and $B$ respectively. Arrange the elements of $A\cup B$ in ascending order to form a new sequence $\{c_{n}\}$. Find $T_{20}=c_{1}+c_{2}+c_{3}+\cdots +c_{20}$.

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