
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 $\vec{a} =\left(\cos \frac{3x}{2},\sin \frac{3x}{2}\right)$ and $\vec{b} =\left(\cos \frac{x}{2},\sin - \frac{x}{2}\right)$, with $x\in[0,\frac{\pi}{2}]$.

$(1)$ Find the value of $\vec{a} \cdot \vec{b}$ and $|\vec{a} +\vec{b} |$;

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

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