
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 $y=f(x)=2\sqrt{3}\sin x\cos x+2\cos^2x+a$ ($x\in \mathbb{R}$), where $a$ is a constant.

$(1)$ Find the period of the function $y=f(x)$;

$(2)$ If the minimum value of $y=f(x)$ is $0$, find the value of $a$, and then find the maximum value of $f(x)$ and the equation of the axis of symmetry of the graph.

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