
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 the function $f(x)=2\cos\frac{ω}{2}x\sin(\frac{ω}{2}x-\frac{π}{3})+\frac{\sqrt{3}}{2}, ω>0$, where the distance between two adjacent symmetric axes of the graph of $f(x)$ is $\frac{π}{2}$.  
$(1)$ Find the interval where $f(x)$ is monotonically decreasing.  
$(2)$ If $f(\frac{θ}{2})=-\frac{3}{5}$ and $θ∈[-\frac{π}{6},\frac{5π}{6}]$, find the value of $\sin(θ-\frac{5π}{6})$.

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