
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 $\overrightarrow {b}$, where $|\overrightarrow {a}|=1$, $\overrightarrow {b} = (1,0)$, and $\overrightarrow {a} \perp (\overrightarrow {a} - 2 \overrightarrow {b})$:

1. Find the dot product $\overrightarrow {a} \cdot \overrightarrow {b}$;
2. Find the norm $| \overrightarrow {a} + \overrightarrow {b} |$.

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