
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 function $f(x)=4\cos x \cdot \cos(x-\frac{\pi}{3})+a$.  
$(I)$ Find the smallest positive period of the function $f(x)$.  
$(II)$ Determine the interval of monotonic increase of the function $f(x)$ for $x \in [0,\frac{\pi}{2}]$.  
$(III)$ If $\frac{2\pi}{3}$ is a zero of the function $f(x)$, find the value of the real number $a$ and the range of the function $f(x)$ for $x \in [0,\frac{\pi}{2}]$.

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