
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 $f(x)=\cos^2 \omega x + \sqrt{3}\sin \omega x\cos \omega x - \frac{1}{2}$ ($\omega > 0$), its smallest positive period is $\pi$.

(1) Find the value of $\omega$ and the intervals where $f(x)$ is monotonically increasing;

(2) In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite angles $A$, $B$, and $C$, respectively. Given that $a=1$, $b= \sqrt{2}$, and $f\left( \frac{A}{2}\right) = \frac{\sqrt{3}}{2}$, find the measure of angle $C$.

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