
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 that the function $f(x) = \frac{1}{2}\sin(\omega x + \varphi)$ ($\omega > 0, 0 < \varphi < \pi$) is an even function, and points P and Q are the highest and lowest points respectively on the graph of $y = f(x)$ such that $| \overrightarrow{PQ} | = \sqrt{2}$,  
(1) Find the explicit formula of the function $f(x)$;  
(2) In triangle $\triangle ABC$, where $a, b, c$ are the sides opposite to angles $A, B, C$ respectively, given that $a=1, b= \sqrt{2}$, and $f\left(\frac{A}{\pi}\right) = \frac{\sqrt{3}}{4}$, determine the size of angle $C$.

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