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

1. Find $\overrightarrow{a} \cdot \overrightarrow{b}$ and $| \overrightarrow{a}+ \overrightarrow{b}|$;
2. If the minimum value of $f(x)$ is $- \frac {3}{2}$, find the value of $\lambda$.

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