
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 that the function $f(x)= \sqrt {3}\sin 2ωx-2\sin ^{2}ωx$ has a minimum positive period of $3π$.
$(1)$ Find the analytic expression of the function $f(x)$;
$(2)$ In $\triangle ABC$, if $f(C)=1$, $AB=2$, $2\sin ^{2}B=\cos B+\cos (A-C)$, find the length of $BC$.

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