
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 two vectors $\overrightarrow{a} = (\sqrt{3}\sin x, -1)$ and $\overrightarrow{b} = (\cos x, m)$, where $m \in \mathbb{R}$,

(1) If $m = \tan \frac{10\pi}{3}$ and $\overrightarrow{a} \| \overrightarrow{b}$, find the value of $\cos^2 x - \sin 2x$;

(2) If we translate the graph of the function $f(x) = 2(\overrightarrow{a} + \overrightarrow{b}) \cdot \overrightarrow{b} - 2m^2 - 1$ to the right by $\frac{\pi}{6}$ units to obtain the graph of function $g(x)$, and if $g(x)$ has a zero point in the interval $[0, \frac{\pi}{2}]$, find the range of values for $m$.

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