
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 $\sin \alpha + \cos \alpha = -\frac{1}{5}$.

$(1)$ Find the value of $\sin \left( \frac{\pi}{2} + \alpha \right) \cos \left( \frac{\pi}{2} - \alpha \right)$;  
$(2)$ If $\frac{\pi}{2} < \alpha < \pi$, find the value of $\frac{1}{\sin (\pi - \alpha)} + \frac{1}{\cos (\pi - \alpha)}$.

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