
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 $f(x)=\cos \left( \frac {\pi}{3}+x\right)\cos \left( \frac {\pi}{3}-x\right)$, $g(x)= \frac {1}{2}\sin 2x- \frac {1}{4}$.
$(1)$ Find the smallest positive period of the function $f(x)$;
$(2)$ Find the maximum value of the function $h(x)=f(x)-g(x)$, and the set of $x$ values for which $h(x)$ attains its maximum value.

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