
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
(1) Calculate $\left[(1+2i)\cdot i^{100}+\left(\frac{1-i}{1+i}\right)^{5}\right]^{2}-\left(\frac{1+i}{\sqrt{2}}\right)^{20}$

(2) Given complex numbers $z$ and $ω$, $(1+3i)\cdot z$ is a purely imaginary number, $ω=\frac{z}{2+i}$, and $|ω|=5\sqrt{2}$, find the complex number $z$.

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