
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 that the maximum value of the function $f(x)=2\cos ^{2}x+2 \sqrt {3}\sin x\cos x+a$ is $2$.
$(1)$ Find the value of $a$ and the equation of the axis of symmetry of the graph of $f(x)$;
$(2)$ Translate the graph of $y=f(x)$ to the right by $\frac {\pi}{12}$ units to obtain the graph of $y=g(x)$. Find the range of $g(x)$ on the interval $[\frac {\pi}{6}, \frac {\pi}{3}]$.

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