
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 vectors $\overrightarrow{a}=(\sqrt{3}\sin \omega x, \sin \omega x)$ and $\overrightarrow{b}=(\cos \omega x, \sin \omega x)$, where $\omega > 0$, let the function $f(x)= \overrightarrow{a} \cdot \overrightarrow{b} - \frac{1}{2}$. The distance between two adjacent axes of symmetry of the graph of $f(x)$ is $\frac{\pi}{2}$.

(Ⅰ) Find the expression for $f(x)$ and the intervals where $f(x)$ is monotonically increasing;

(Ⅱ) Suppose in triangle $ABC$, the sides corresponding to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $a+b=3$, $c=\sqrt{3}$, and $f(C)=1$, find the area of triangle $ABC$.

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