
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 $\cos(x-\frac{π}{4})=\frac{\sqrt{2}}{10}$, $x∈(\frac{π}{2},\frac{3π}{4})$.  
$(1)$ Find the value of $\cos x$;  
$(2)$ Find the value of $\sin(2x+\frac{π}{3})$.

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