
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 \(a\), \(b\), \(c\), and \(d\) be distinct, nonzero real numbers such that
\[
a^2 + \frac{1}{b} = b^2 + \frac{1}{c} = c^2 + \frac{1}{d} = d^2 + \frac{1}{a}.
\]
Find \(|abcd|\).

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