
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 $\sin \alpha - \cos \alpha = \frac{\sqrt{10}}{5}$ and $\alpha$ is in the interval $(\pi, 2\pi)$:  
$(1)$ Find the value of $\sin \alpha \cos \alpha$;  
$(2)$ Find the value of $\sin \alpha + \cos \alpha$.

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