
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 vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ satisfying $|\overrightarrow {a}|= \sqrt {2}$, $|\overrightarrow {b}|=2$, and $(\overrightarrow {a} - \overrightarrow {b}) \perp \overrightarrow {a}$, find the following:
- The dot product $\overrightarrow {a} \cdot \overrightarrow {b} = \_\_\_\_\_\_$;
- The angle between the vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ equals $\_\_\_\_\_\_$;
- The magnitude of $2\overrightarrow {a} - \overrightarrow {b} = \_\_\_\_\_\_$.

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