
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 $\overrightarrow {a} = (\cos \frac {3}{2}x, \sin \frac {3}{2}x)$ and $\overrightarrow {b} = (\cos \frac {x}{2}, -\sin \frac {x}{2})$, with $x \in \left[-\frac {\pi}{3}, \frac {\pi}{4}\right]$,  
(1) Find $\overrightarrow {a} \cdot \overrightarrow {b}$ and $|\overrightarrow {a} + \overrightarrow {b}|$;  
(2) If $f(x) = \overrightarrow {a} \cdot \overrightarrow {b} - |\overrightarrow {a} + \overrightarrow {b}|$, find the maximum and minimum values of $f(x)$.

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