
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
Find the set of all real numbers \(x\) for which 
\[ x^2 - \sqrt{x^2 + 1} - \frac{1}{x^2 - \sqrt{x^2 + 1}} \]
is a rational number.
A) \(x = 0\)  
B) \(x \in \mathbb{Q}\)  
C) All real \(x\)  
D) \(x\) for which \(x^2 + 1\) is rational  
E) \(x\) for which \(x^2 - \sqrt{x^2 + 1}\) is rational

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