
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
Let  $a \geq 0$  be a natural number. Determine all rational  $x$ , so that
\[\sqrt{1+(a-1)\sqrt[3]x}=\sqrt{1+(a-1)\sqrt x}\]
All occurring square roots, are not negative.**Note.** It seems the set of natural numbers =  $\mathbb N = \{0,1,2,\ldots\}$  in this problem.

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