
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
For a geometric sequence $\{a_n\}$ where each term is positive and satisfies $a_1a_7=4$ and $a_6=8$, if the function $f(x)=a_1x+a_2x^2+a_3x^3+\ldots+a_{10}x^{10}$ has a derivative $f′(x)$, then $f′\left( \frac{1}{2}\right)=$ \_\_\_\_\_\_.

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