
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
Consider the set of all real numbers for which:
\[
x + \sqrt{x^2+4x} - \frac{1}{x+\sqrt{x^2+4x}}
\]
is a rational number. Identify the set of all such $x$.

####
A) integers $x$  
B) rational $x$  
C) real $x$  
D) $x$ for which $\sqrt{4 + x^2}$ is rational  
E) $x$ for which $x + \sqrt{x^2 + 4x}$ is rational

####

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