
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 planar vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy $(\overrightarrow{a} + \overrightarrow{b}) \cdot (2\overrightarrow{a} - \overrightarrow{b}) = -12$, and $|\overrightarrow{a}| = 2$, $|\overrightarrow{b}| = 4$, find the projection of $\overrightarrow{b}$ onto $\overrightarrow{a}$.

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