
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 $\overrightarrow {a}$ and $\overrightarrow {b}$ that satisfy: $|\overrightarrow {a}| = \sqrt {2}$, $|\overrightarrow {b}| = 4$, and $\overrightarrow {a} \cdot (\overrightarrow {b} - \overrightarrow {a}) = 2$.

1. Find the angle between vectors $\overrightarrow {a}$ and $\overrightarrow {b}$.
2. Find the minimum value of $|t \overrightarrow {a} - \overrightarrow {b}|$ and the value of $t$ when the minimum value is attained.

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