
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
Let $a, b, c, d$ be real numbers. Given that the roots of the polynomial $z^4 + 2az^3 + 3bz^2 + 4cz + 5d = 0$ are complex numbers that lie on a circle in the complex plane centered at $0+0i$ with radius 1, determine the sum of the reciprocals of the roots.

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