
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
Let the function $f(x)=\cos (2x- \frac {\pi}{6})-2\sin x\cos x$.  
$(1)$ Find the interval of monotonic decrease for $f(x)$;  
$(2)$ In $\triangle ABC$, if $AB=4$ and $f( \frac {C}{2})= \frac {1}{2}$, find the area of the circumcircle of $\triangle ABC$.

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