
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 $$\sqrt {2}\sin \frac {x}{4}\cos \frac {x}{4}+ \sqrt {6}\cos^2 \frac {x}{4}- \frac { \sqrt {6}}{2}-m\ge0$$ holds for all $$x \in \left[-\frac {\pi}{3}, \frac {\pi}{3}\right]$$, determine the range of the real number $m$.

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