
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 $\overrightarrow{m}=( \sqrt{3}\sin \frac{x}{4},1)$, $\overrightarrow{n}=(\cos \frac{x}{4},\cos \frac{2x}{4})$, let $f(x)=\overrightarrow{m}\cdot \overrightarrow{n}$.  
(Ⅰ) Find the interval of monotonic decrease for $f(x)$;  
(Ⅱ) If the graph of the function $y=f(x)$ is translated to the right by $\frac{2\pi}{3}$ units to obtain the graph of $y=g(x)$, and if the function $y=g(x)-k$ has a zero in the interval $\left[0, \frac{7\pi}{3}\right]$, find the range of the real number $k$.

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