
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
When $x \in \left[-\frac{\pi}{3}, \frac{\pi}{3}\right]$, find the minimum value of the function $f(x) = \sqrt{2}\sin \frac{x}{4}\cos \frac{x}{4} + \sqrt{6}\cos^2 \frac{x}{4} - \frac{\sqrt{6}}{2}$.

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