
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 $α \in (0, \frac{π}{4}), β \in (\frac{π}{4}, \frac{π}{2}), \vec{a}=(2\cos^2α,2\sin α\cos α)$, $\vec{b}=(1+2\sin β\cos β,1-2\sin^2β)$, $\vec{c}=(1,0)$. Let $<\vec{a}, \vec{c}> = θ_1, <\vec{b}, \vec{c}> = θ_2$.

$(1)$ If $θ_2= \frac{π}{6}$, find the angle $β$;

$(2)$ If $θ_2-θ_1= \frac{π}{6}$, find $\sin (β-α)$.

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