
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 that $α∈[\dfrac{π}{2}, \dfrac{3π}{2}]$, $β∈[-\dfrac{π}{2}, 0]$, and the equations $(α-\dfrac{π}{2})^{3}-\sin α-2=0$ and $8β^{3}+2\cos^{2}β+1=0$ hold, find the value of $\sin(\dfrac{α}{2}+β)$.

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