
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
Let the function $f(x)=2\cos x(\cos x+\sqrt{3}\sin x)$ $(x\in\mathbb{R})$.

$(1)$ Find the period of the function $y=f(x)$ and its intervals of monotonic increase;

$(2)$ When $x\in\left[0, \frac{\pi}{2}\right]$, find the maximum value of the function $f(x)$.

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