
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 $\overrightarrow{a} = (\cos\left(\frac{3x}{2}\right), \sin\left(\frac{3x}{2}\right))$ and $\overrightarrow{b} = (\cos\left(\frac{x}{2}\right), -\sin\left(\frac{x}{2}\right))$, with $x \in \left[0, \frac{\pi}{2}\right]$.
1. Determine the dot product $\overrightarrow{a} \cdot \overrightarrow{b}$ and the magnitude $|\overrightarrow{a} + \overrightarrow{b}|$.
2. If $f(x) = \overrightarrow{a} \cdot \overrightarrow{b} - 2\lambda|\overrightarrow{a} + \overrightarrow{b}|$ reaches a minimum value of $-2$, find the value of $\lambda$.

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