
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 $f(x)=\sin (\omega x-\varphi)$, where $(\omega > 0, 0 < \varphi < \frac{\pi}{2})$, the graph passes through the point $(\frac{\pi}{4}, \frac{\sqrt{3}}{2})$, and the distance between two adjacent axes of symmetry is $\frac{\pi}{2}$.  
(I) Find the expression for $f(x)$ and its increasing interval on $[0,\pi]$;  
(II) In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively. If $f(\frac{A}{2}) + \cos A = \frac{1}{2}$, find the measure of $\angle A$.

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