
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
Determine all positive integers  $n$  for which there are positive real numbers  $x,y$    and   $z$   such that  $\sqrt x +\sqrt y +\sqrt z=1$  and  $\sqrt{x+n} +\sqrt{y+n} +\sqrt{z+n}$  is an integer. 

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