
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 $0 < \alpha < \frac{\pi}{2} < \beta < \pi$, $\cos \left(\beta - \frac{\pi}{4}\right) = \frac{1}{3}$, and $\sin(\alpha+\beta)= \frac{4}{5}$.

$(1)$ Find the value of $\sin 2\beta$;

$(2)$ Find the value of $\cos\left(\alpha+ \frac{\pi}{4}\right)$.

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