
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 plane vectors $\overrightarrow {a}$, $\overrightarrow {b}$, $\overrightarrow {e}$, where $| \overrightarrow {e}|=1$, $\overrightarrow {a} \cdot \overrightarrow {e} = 2$, $\overrightarrow {b} \cdot \overrightarrow {e} = -1$, and $| \overrightarrow {a} + \overrightarrow {b}|=2$, determine the maximum value of $\overrightarrow {a} \cdot \overrightarrow {b}$.

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