
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 $0 \leqslant \varphi < \pi $, the function is defined as $f(x)=\frac{\sqrt{3}}{2}\cos (2x+ \varphi )+{\sin }^{2}x$.

(I) If $\varphi = \frac{\pi }{6}$, find the interval where $f(x)$ is monotonically increasing.

(II) If the maximum value of $f(x)$ is $\frac{3}{2}$, find the value of $\varphi$.

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