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

$(1)$ The magnitude of $\overrightarrow{a}+\overrightarrow{b}$;

$(2)$ 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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