
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 $\frac{3\pi}{4} < \alpha < \pi$, find the values for the following:
1. $\tan\alpha$ given $\tan\alpha + \frac{1}{\tan\alpha} = -\frac{10}{3}$.
2. The value of $$\frac{5\sin^2\left(\frac{\alpha}{2}\right) + 8\sin\left(\frac{\alpha}{2}\right)\cos\left(\frac{\alpha}{2}\right) + 11\cos^2\left(\frac{\alpha}{2}\right) - 8}{\sqrt{2}\sin\left(\alpha - \frac{\pi}{4}\right)}.$$

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