
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 that $x$ satisfies the inequality $2(\log_{\frac{1}{2}}x)^{2}+3\leqslant \log_{\frac{1}{2}}x^{7}$, find the maximum and minimum values of the function $f(x)=\log_{\frac{1}{2}}(2x)\cdot\log_{\frac{1}{2}}(4x)$ and the corresponding values of $x$.

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