
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:
(I) $\overrightarrow{a} \cdot \overrightarrow{b}$ and $| \overrightarrow{a} + \overrightarrow{b} |$;
(II) 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}
user