
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
Complex numbers $u$, $v$, and $w$ are zeros of the polynomial $Q(z) = z^3 + sz + t$, and $|u|^2 + |v|^2 + |w|^2 = 400$. The points corresponding to $u$, $v$, and $w$ in the complex plane form a right triangle with hypotenuse $k$. Determine $k^2$.

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