
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
For what real values of \( x \) is \( \sqrt{x + \sqrt{2x-1}} + \sqrt{x - \sqrt{2x-1}} = A \), given (a) \( A = \sqrt{2} \), (b) \( A = 1 \), (c) \( A = 2 \), where only non-negative real numbers are allowed in square roots and the root always denotes the non-negative root?

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