
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 $f(x) = 2\sin x\cos^2 \frac{\varphi}{2} + \cos x\sin \varphi - \sin x$ $(0 < \varphi < \pi)$ attains its minimum value at $x=\pi$.

(1) Find the value of $\varphi$;

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

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