
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 $f(x)=2\cos x(\sin x-\sqrt{3}\cos x)+\sqrt{3}$.  
$(Ⅰ)$ Find the minimum positive period of $f(x)$ and the interval where $f(x)$ is monotonically decreasing.  
$(Ⅱ)$ When $x∈[\frac{π}{2},π]$, find the minimum value of the function $f(x)$ and the corresponding value of $x$.

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