
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 $a,$ $b,$ $c,$ and $d$ are zeros of a polynomial $P(z) = z^4 + pz^2 + qz + r,$ and $|a|^2 + |b|^2 + |c|^2 + |d|^2 = 340.$ The points corresponding to $a,$ $b,$ $c,$ and $d$ in the complex plane form a quadrilateral where the diagonal connecting $a$ to $c$ is a segment of the right angle at $b$. Find the square of the length of this diagonal.

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