
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 distinct unit vectors $\overrightarrow{a}$, $\overrightarrow{b}$, and $\overrightarrow{c}$ on a plane that satisfy $\overrightarrow{a} \cdot \overrightarrow{b} = \overrightarrow{b} \cdot \overrightarrow{c} = \frac{1}{2}$, find the maximum value of $| \overrightarrow{a} \cdot \overrightarrow{e} | + | 2 \overrightarrow{b} \cdot \overrightarrow{e} | + 3 | \overrightarrow{c} \cdot \overrightarrow{e} |$, where $\overrightarrow{e}$ is any unit vector on the plane.

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