
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
Points \( (\sqrt{e}, a) \) and \( (\sqrt{e}, b) \) are distinct points on the graph of \( y^2 + x^4 = 3x^2y + 1 \). What is \( |a-b| \)?
**A)** \( \sqrt{3e^2 + 2} \)
**B)** \( 2\sqrt{e} \)
**C)** \( \sqrt{5e^2 + 4} \)
**D)** \( \sqrt{7} \)

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