
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 cross product of vectors as
$$
|\boldsymbol{a} \times \boldsymbol{b}|=|\boldsymbol{a}||\boldsymbol{b}| \sin \langle\boldsymbol{a}, \boldsymbol{b}\rangle,
$$

where $\langle\boldsymbol{a}, \boldsymbol{b}\rangle$ indicates the angle between vectors $\boldsymbol{a}$ and $\boldsymbol{b}$. Given three vectors $\boldsymbol{a}$, $\boldsymbol{b}$, and $\boldsymbol{c}$ such that $\boldsymbol{a} + \boldsymbol{b} + \boldsymbol{c} = \mathbf{0}$ and their magnitudes are equal, find the range of values for
$$
\left|\frac{|a \times b + b \times c + c \times a|}{a \cdot b + b \cdot c + c \cdot a}\right|
$$

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