
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
Define the "companion function" of a non-zero vector $\overrightarrow {OM} = (a, b)$ as $f(x) = a\sin x + b\cos x$ ($x \in \mathbb{R}$), and the vector $\overrightarrow {OM} = (a, b)$ is called the "companion vector" of the function $f(x) = a\sin x + b\cos x$ ($x \in \mathbb{R}$), where $O$ is the origin of coordinates. Denote the set of all vectors' "companion functions" in the plane as $S$.  
(1) Let $h(x) = \sqrt{3}\cos(x + \frac{\pi}{6}) + 3\cos(\frac{\pi}{3} - x)$ ($x \in \mathbb{R}$), please determine whether the function $h(x)$ has a companion vector $\overrightarrow {OM}$. If it exists, find the collinear unit vector with $\overrightarrow {OM}$; if not, please explain why.  
(2) Given a point $M(a, b)$ satisfying $\frac{b}{a} \in (0, \sqrt{3}]$, the "companion function" $f(x)$ of the vector $\overrightarrow {OM}$ attains its maximum value at $x = x_0$, find the range of values for $\tan 2x_0$.

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