
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 vectors $\overrightarrow {a} = (\sqrt {3}\sin x, -1)$ and $\overrightarrow {b} = (\cos x, m)$, where $m \in \mathbb{R}$.
(1) If $m = \sqrt {3}$ and $\overrightarrow {a} \parallel \overrightarrow {b}$, find the value of $\frac{3 \sin x - \cos x}{\sin x + \cos x}$.
(2) Given the function $f(x) = 2(\overrightarrow {a} + \overrightarrow {b}) \cdot \overrightarrow {b} - 2m^2 -1$. If function $f(x)$ has a zero 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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