
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 $a_{n}+a_{n-1}=8n+2\left(n\geqslant 2\right)$, and a geometric sequence $\{b_{n}\}$ with a common ratio of $3$, where $a_{2}+b_{2}=20$.<br/>$(1)$ Find the general formulas for sequences $\{a_{n}\}$ and $\{b_{n}\}$;<br/>$(2)$ The terms of sequences $\{a_{n}\}$ and $\{b_{n}\}$ are combined in ascending order to form a new sequence $\{c_{n}\}$. Let $S_{n}$ denote the sum of the first $n$ terms of sequence $\{c_{n}\}$. Find $S_{50}$.

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