
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 non-zero vectors $\vec{a}$ and $\vec{b}$ satisfy $|\vec{a}|=\sqrt{3}|\vec{b}|$, if the function $f(x)= \frac{1}{3}x^{3}+|\vec{a}|x^{2}+2\vec{a}\cdot\vec{b}x+1$ has an extremum for $x\in\mathbb{R}$, then the range of values for $<\vec{a},\vec{b}>$ is \_\_\_\_\_\_\_\_.

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