
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 all positive integers  $n$  for which there exist real numbers  $x_1, x_2,. . . , x_n$  satisfying all of the following conditions: 
(i)  $-1 <x_i <1,$  for all  $1\leq  i \leq  n.$ (ii)  $ x_1 + x_2 + ... + x_n = 0.$ (iii)  $\sqrt{1 - x_1^2} +\sqrt{1 - x^2_2} + ... +\sqrt{1 - x^2_n} = 1.$ 

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