
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 $f(x)=2\sin x \cdot \cos^{2}\frac{\theta}{2} + \cos x \cdot \sin\theta - \sin x$ $(0 < \theta < \pi)$ takes its minimum value at $x=\pi$.  
$(1)$ Find the value of $\theta$;  
$(2)$ In $\triangle ABC$, $a, b, c$ are the sides opposite to angles $A, B, C$ respectively, given that $a=1, b=\sqrt{2}, f(A)=\frac{\sqrt{3}}{2}$, find the angle $C$.

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