
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 $f(x)=( \sqrt {3}\sin \omega x+\cos \omega x)\cos \omega x- \frac {1}{2}(x\in\mathbb{R},\omega > 0)$. If the smallest positive period of $f(x)$ is $4\pi$.  
$(1)$ Find the intervals of monotonic increase for the function $f(x)$;  
$(2)$ In $\triangle ABC$, the sides opposite angles $A$, $B$, $C$ are respectively $a$, $b$, $c$, and it satisfies $(2a-c)\cos B=b\cos C$, find the range of values for $f(A)$.

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