
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 $\overrightarrow{a}=(\cos \frac {3x}{2},\sin \frac {3x}{2})$ and $\overrightarrow{b}=(\cos \frac {x}{2},-\sin \frac {x}{2})$, where $x\in[-\frac {\pi}{3}, \frac {\pi}{4}]$.

(I) Find $\overrightarrow{a}\cdot \overrightarrow{b}$ and $|\overrightarrow{a}+ \overrightarrow{b}|$.

(II) 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}
user