
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
The product of the first $n$ terms of the sequence $\{a_{n}\}$ is $b_{n}$, where $b_{1}=1$.<br/>$(1)$ If $b_{n+1}=2a_{n}$, find $b_{4}$;<br/>$(2)$ If $b_n=2^\frac{n(n-1)}{2}$, and $c_{n}=\log _{4}a_{2n-1}$, find the sum of the first $n$ terms of the sequence $\{c_{n}\}$.

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