
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 $\sin \alpha= \frac{4}{5}$, with $\alpha \in \left( \frac{\pi}{2},\pi\right)$, and $\cos \beta= -\frac{5}{13}$, where $\beta$ is in the third quadrant.

$(1)$ Find the value of $\sin (\alpha-\beta)$  
$(2)$ Find the value of $\tan (\alpha+\beta)$.

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