
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 $f(x)=\cos ^{2}ωx-\sin ^{2}ωx+2 \sqrt {3}\cos ωx\cdot\sin ωx$, where $ω > 0$. If the distance between two adjacent symmetry axes of $f(x)$ is not less than $\frac {π}{2}$
(1) Find the range of $ω$ and the interval where the function $f(x)$ is monotonically increasing;
(2) In $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively, with $a= \sqrt {3}$, $b+c=3$. When $ω$ 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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