
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 $f(x)= \frac {1}{2}\cos ^{2}x+ \frac { \sqrt {3}}{2}\sin x\cos x+1$, where $x\in\mathbb{R}$.  
$(1)$ Find the smallest positive period of the function $f(x)$.  
$(2)$ Find the maximum and minimum values of the function $f(x)$ in the interval $\left[ \frac {\pi}{12}, \frac {\pi}{4}\right]$ and determine the values of the variable $x$ when the function attains its maximum and minimum values.

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