
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
Write the complex number \( z=-\sqrt{2} i\left(\cos \frac{3 \pi}{4}+i \sin \frac{3 \pi}{4}\right) \) in algebraic and trigonometric forms.

Perform the multiplication:

\( 3\left(\cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}\right) \cdot \frac{1}{6}\left(\cos \frac{5 \pi}{6}+i \sin \frac{5 \pi}{6}\right) \).

\( (4-4 i) \cdot 3 \sqrt{2}\left(\cos 75^{\circ}+i \sin 75^{\circ}\right) \).

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