
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 vectors $\overrightarrow{m}=(\sqrt{3}\sin \frac{x}{4},1)$, $\overrightarrow{n}=(\cos \frac{x}{4},\cos^2 \frac{x}{4})$. If $\overrightarrow{m} \perp \overrightarrow{n}$, then the value of $\cos\left( x+\frac{\pi}{3} \right)$ is ______.

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