
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 $f(x)=\frac{1}{2}(sinx+cosx)(sinx-cosx)+\sqrt{3}sinxcosx$. Find:<br/>
$(1)$ The interval where the function $f(x)$ is monotonically increasing;<br/>
$(2)$ If $f(\frac{α}{2}+\frac{π}{4})=\frac{\sqrt{3}}{3}$ and $-\frac{π}{2}＜α＜0$, find the value of $\cos \alpha$.

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