
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 a sequence $\{a_{n}\}$ where $a_{1}=1$ and $a_{n+1}=\left\{\begin{array}{l}{{a}_{n}+1, n \text{ is odd}}\\{{a}_{n}+2, n \text{ is even}}\end{array}\right.$

$(1)$ Let $b_{n}=a_{2n}$, write down $b_{1}$ and $b_{2}$, and find the general formula for the sequence $\{b_{n}\}$.

$(2)$ Find the sum of the first $20$ terms of the sequence $\{a_{n}\}$.

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