
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 $\tan 2\theta = -2\sqrt{2}$, and $\pi < 2\theta < 2\pi$.

(Ⅰ) Find the value of $\tan \theta$;  
(Ⅱ) Calculate the value of $\frac{2\cos^2 \frac{\theta}{2} - \sin \theta - 1}{\sqrt{2}\sin\left(\theta + \frac{\pi}{4}\right)}$.

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