
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 the space vectors $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ that satisfy $\overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c} = \overrightarrow{0}$, and $|\overrightarrow{a}| = 3, |\overrightarrow{b}| = 1, |\overrightarrow{c}| = 4$, find the value of $\overrightarrow{a} \cdot \overrightarrow{b} + \overrightarrow{b} \cdot \overrightarrow{c} + \overrightarrow{a} \cdot \overrightarrow{c}$ .

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