
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 $\overrightarrow{a}=(\cos \frac{3}{2}x,\sin \frac{3}{2}x)$, $\overrightarrow{b}=(\cos \frac{x}{2},\sin \frac{x}{2})$, where $x\in[0, \frac{\pi}{2}]$, and $f(x)=\overrightarrow{a}\cdot \overrightarrow{b}-2\lambda|\overrightarrow{a}-\overrightarrow{b}|$ ($\lambda$ is a constant), find:

1. $\overrightarrow{a}\cdot \overrightarrow{b}$ and $|\overrightarrow{a}-\overrightarrow{b}|$;
2. The value of the real number $\lambda$ if the maximum value of $f(x)$ is $\frac{3}{2}$.

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