
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 two plane vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ satisfying $|\overrightarrow {a}|=\sqrt {2}$ and $|\overrightarrow {b}|=1$,
(1) If $|\overrightarrow {a}-\overrightarrow {b}|=2$, find the cosine of the angle between $\overrightarrow {a}$ and $\overrightarrow {b}$.
(2) If for all real numbers $x$, $|\overrightarrow {a}+x\overrightarrow {b}|\geq|\overrightarrow {a}+\overrightarrow {b}|$ is always true, find the angle between $\overrightarrow {a}$ and $\overrightarrow {b}$.

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