
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 $f(x)=\sqrt{3}\sin \omega x-\cos \omega x+m (\omega > 0, x\in R, m$ is a constant$)$ whose graph has a highest point at $(\frac{\pi }{3},1)$ and the closest lowest point at $(-\frac{\pi }{6},-3)$.

(I) Find the analytical expression of the function $f(x)$ and its monotonically increasing interval;

(II) In $\Delta ABC$, the sides opposite to angles $A, B, C$ are $a, b, c$ respectively, and $\overrightarrow{AB}\cdot \overrightarrow{BC}=-\frac{1}{2}ac$, find the range of the function $f(A)$.

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