
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]


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Given the function $f(x) = \cos^2\omega x - \sin^2\omega x + 2\sqrt{3}\cos\omega x\sin\omega x$, where $\omega > 0$, if the distance between two adjacent axes of symmetry of $f(x)$ is not less than $\frac{\pi}{2}$,  
(1) find the range of $\omega$ and the intervals of monotonic increase for the function $f(x)$;  
(2) In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite angles $A$, $B$, and $C$ respectively, with $a= \sqrt{3}$, $b+c=3$. When $\omega$ is at its maximum, $f(A) = 1$, find the value of $\sin B \cdot \sin C$.

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