
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 $f\left( x \right)=\sin \left( \omega x-\varphi \right)$, where $\omega > 0$ and $0 < \varphi < \frac{\pi }{2}$, its graph passes through the point $\left( \frac{\pi}{4}, \frac{ \sqrt{3}}{2}\right)$, and the distance between adjacent symmetry axes is $\frac{\pi}{2}$.

(I) Find the analytical expression of the function $f\left(x\right)$ and its monotonically increasing interval within $\left[0,\pi\right]$;

(II) In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$, respectively. If $f\left( \frac{A}{2} \right)+\cos A=\frac{1}{2}$, find the magnitude of $\angle A$.

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