
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 vectors $\overrightarrow{a}$, $\overrightarrow{b}$, and $\overrightarrow{c}$ satisfying $\overrightarrow{a} + \overrightarrow{b} + 2\overrightarrow{c} = \overrightarrow{0}$, and the magnitudes $|\overrightarrow{a}|=1$, $|\overrightarrow{b}|=3$, $|\overrightarrow{c}|=2$, find the value of $\overrightarrow{a} \cdot \overrightarrow{b} + 2\overrightarrow{a} \cdot \overrightarrow{c} + 2\overrightarrow{b} \cdot \overrightarrow{c}$.

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