
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 $f(x)=\sin ωx \cdot \cos ωx - \frac{ \sqrt{3}}{2} + \sqrt{3}\cos ^{2}ωx (ω > 0)$, with its smallest positive period being $π$.
(I) Find the interval where $f(x)$ is monotonically increasing;
(II) Let $a$, $b$, and $c$ be the sides opposite to angles $A$, $B$, and $C$ in $\triangle ABC$, respectively. If angle $A$ is acute, $f(A)=0$, $a=1$, and $b+c=2$, find the area of $\triangle ABC$.

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