
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]


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Given that points \(A(\sqrt{\pi}, a)\) and \(B(\sqrt{\pi}, b)\) are two distinct points on the curve \(y^{2} + x^{4} = 2x^{2}y + 1\), find the value of \(|a - b|\).

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