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

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