
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 $f(x)=2\cos ^{2} \frac {ωx}{2}+\cos (ωx+ \frac {π}{3})$, where the smallest positive period of $ω$ is $π$.
(I) Find the value of $ω$ and the monotonically decreasing interval of the function $f(x)$;
(II) In acute triangle $ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively. If $f(A)=- \frac {1}{2}$, $c=3$, and the area of $△ABC$ is $6 \sqrt {3}$, find the area of the circumcircle of $△ABC$.

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