
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 $\overrightarrow{a}=(\cos \frac{3}{2}x,\sin \frac{3}{2}x)$, $\overrightarrow{b}=(\cos \frac{x}{2},-\sin \frac{x}{2})$, and $x\in[0,\frac{\pi }{2}]$. Find:

$(1)$ $\overrightarrow{a}\cdot \overrightarrow{b}$ and $|\overrightarrow{a}+\overrightarrow{b}|$;

$(2)$ If the minimum value of $f(x)=\overrightarrow{a}\cdot \overrightarrow{b}-2\lambda |\overrightarrow{a}+\overrightarrow{b}|$ is $(-\frac{3}{2})$, find the value of the real number $\lambda$.

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