
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
Let $f(x)=2\sin x\cos x-2\cos ^{2}(x+\frac{π}{4})$.  
$(1)$ Find the intervals where $f(x)$ is monotonically increasing and its center of symmetry.  
$(2)$ Given $x\in (0,\frac{π}{2})$, if $f(x+\frac{π}{6})=\frac{3}{5}$, find the value of $\cos 2x$.

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