
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 three different unit vectors $\overrightarrow{a}$, $\overrightarrow{b}$, $\overrightarrow{c}$ on a plane satisfying $\overrightarrow{a} \cdot \overrightarrow{b} = \overrightarrow{b} \cdot \overrightarrow{c} = \frac{1}{2}$, if $\overrightarrow{e}$ is any unit vector in the plane, then the maximum value of $|\overrightarrow{a} \cdot \overrightarrow{e}| + |2\overrightarrow{b} \cdot \overrightarrow{e}| + 3|\overrightarrow{c} \cdot \overrightarrow{e}|$ is \_\_\_\_\_\_.

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