
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 $\overrightarrow {a} = (\sin x + \cos x, \sqrt {2}\cos x)$ and $\overrightarrow {b} = (\cos x - \sin x, \sqrt {2}\sin x)$, where $x \in \left[-\frac{\pi}{8}, 0\right]$.  
(1) Find the range of $|\overrightarrow {a}|$.  
(2) Find the range of the function $f(x) = \overrightarrow {a} \cdot \overrightarrow {b} - |\overrightarrow {a}|$.

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