
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 $ \tan \left( \alpha + \frac{\pi}{4} \right) = -\frac{1}{2} $ and $\frac{\pi}{2} < \alpha < \pi$, calculate the value of $ \frac{\sin 2\alpha - 2\cos^2 \alpha}{\sin \left(\alpha - \frac{\pi}{4} \right)} $.

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