
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 the inequality $f(x) = 3\sqrt{2}\sin\left(\frac{x}{4}\right)\cos\left(\frac{x}{4}\right) + \sqrt{6}\cos^2\left(\frac{x}{4}\right) - \frac{\sqrt{6}}{2} + m \leq 0$, determine the range of the real number $m$ such that the inequality holds for any $-\frac{5\pi}{6} \leq x \leq \frac{\pi}{6}$.

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