
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 the sequence $\{a_{n}\}$ satisfy: ${a}_{n+1}=\{\begin{array}{l}2{a}_{n},n=2k\\{a}_{n}+1,n=2k-1\end{array}(k∈{N}^{*}),{a}_{2}$ is the geometric mean of $a_{1}$ and $a_{3}$. <br/>$(1)$ Find the value of $a_{1}$; <br/>$(2)$ Find the sum of the first $20$ terms of the sequence $\{a_{n}\}$.

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