
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 $\overrightarrow{m} = \left(\sqrt{3}\sin\frac{x}{4}, 1\right)$, $\overrightarrow{n} = \left(\cos\frac{x}{4}, \cos^2\frac{x}{4}\right)$, let $f(x) = \overrightarrow{m} \cdot \overrightarrow{n}$.
(1) Find the smallest positive period and the interval of monotonic increase for the function $f(x)$;
(2) The graph of function $y=f(x)$ is translated to the right by $\frac{2\pi}{3}$ units to obtain the graph of $y=g(x)$, discuss the number of zeros of the function $y=g(x)-k$ on the interval $\left[0, \frac{7\pi}{3}\right]$.

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