
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 $x \in [\frac{\pi}{6}, \frac{2\pi}{3})$:

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 range of $f(x)$.

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