
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
Let $f(x)$ be the middle term in the expansion of $({x^2+\frac{1}{2x}})^6$. If $f(x) \leq mx$ holds true in the interval $[\frac{\sqrt{2}}{2}, \sqrt{2}]$, then the range of values for the real number $m$ is ___.

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