
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 $y=4\cos^2x-4\sqrt{3}\sin x\cos x-1$ ($x\in\mathbb{R}$).

$(1)$ Find the smallest positive period of the function;  
$(2)$ Find the maximum value of the function and the corresponding $x$ values;  
$(3)$ Find the intervals of monotonic increase;  
$(4)$ Find the axis of symmetry of the function.

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