
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 $x\in(0, \frac{\pi}{2})$.
$(1)$ Find $\overrightarrow{a} \cdot \overrightarrow{b}$ and $|\overrightarrow{a}+ \overrightarrow{b}|$;
$(2)$ If $f(x)=\overrightarrow{a} \cdot \overrightarrow{b} - 2\lambda|\overrightarrow{a}+ \overrightarrow{b}|$ has a minimum value of $- \frac{3}{2}$, find the value of the positive real number $\lambda$.

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