
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 $\overrightarrow{b}=(\cos \frac{1}{2}x,-\sin \frac{1}{2}x)$, where $x\in[0,\frac{\pi}{2}]$, find:
1. The dot product $\overrightarrow{a}\cdot\overrightarrow{b}$ and the magnitude $|\overrightarrow{a}+\overrightarrow{b}|$;
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 $\lambda$.

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