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

$(1)$ Find the expression of $f(x)$;

$(2)$ Let $\triangle ABC$ have angles $A$, $B$, $C$ and corresponding sides $a$, $b$, $c$ respectively, with $a+b=3$, $c=\sqrt{3}$, and $f(C)=1$. Find the area of $\triangle ABC$.

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