
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
Positive sequences  $\{a_n\},\{b_n\}$  satisfy: $a_1=b_1=1,b_n=a_nb_{n-1}-\frac{1}{4}(n\geq 2)$ .
Find the minimum value of  $4\sqrt{b_1b_2\cdots b_m}+\sum_{k=1}^m\frac{1}{a_1a_2\cdots a_k}$ ,where  $m$  is a given positive integer.

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