
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 positive geometric sequence $\{a_n\}$, it is known that $a_1=4$ and $a_5^2=16a_2 \cdot a_6$, find the value of $\dfrac{1}{\sqrt{a_1}} + \dfrac{2}{\sqrt{a_2}} + \dfrac{3}{\sqrt{a_3}} + \ldots + \dfrac{n}{\sqrt{a_n}}$.

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