
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 $\{a_{n}\}$, $\{b_{n}\}$ satisfies ${a_{n-1}}=\left\{{\begin{array}{l}{{b_{\frac{n}{2}}},n=2k,}\\{\sqrt{{b_n}},n=2k+1,}\end{array}k∈{{N}_+}}\right.$, and $\{a_{n}\}$ is a geometric sequence with a common ratio of $2$. Find $\frac{{{b_{n+1}}}}{{{b_n}}}$ and $a_{1}$.

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