
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 the following trigonometric equation:
$$
\frac{\operatorname{tg}(\pi+2 \alpha) \operatorname{ct}\left(\frac{3}{2} \pi+\alpha\right)}{\operatorname{tg} 2 \alpha-\operatorname{tg} \alpha}+2 \cos \left(\frac{\pi}{4}-\alpha\right) \cos \left(\frac{\pi}{4}+\alpha\right)=\sqrt{2} \sin \left(\frac{\pi}{4}-2 \alpha\right),
$$
simplify and verify the equality.

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