
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 vectors $\overrightarrow{m} = (\cos\frac{x}{2} - 1)$ and $\overrightarrow{n} = \left(\sqrt{3}\sin\frac{x}{2}, \cos^2\frac{x}{2}\right)$, and the function $f(x) = \overrightarrow{m} \cdot \overrightarrow{n} + 1$.
(I) If $x \in \left[\frac{\pi}{2}, \pi\right]$, find the minimum value of $f(x)$ and the corresponding value of $x$;
(II) If $x \in [0, \frac{\pi}{2}]$ and $f(x) = \frac{11}{10}$, find the value of $\sin x$.

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