
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 $f(x)=2\cos ^{2} \frac {ωx}{2}+\cos (ωx+ \frac {π}{3})$ with a minimum positive period of $π$ (where $ω > 0$), and $a$, $b$, and $c$ are the sides opposite angles $A$, $B$, and $C$, respectively, in acute triangle $△ABC$, if $f(A)=- \frac {1}{2}$, $c=3$, and the area of $△ABC$ is $6 \sqrt {3}$, determine the area of the circumcircle of $△ABC$.

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