
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
We say that a quadruple of nonnegative real numbers  $(a,b,c,d)$  is *balanced*if   $$ a+b+c+d=a^2+b^2+c^2+d^2. $$      Find all positive real numbers  $x$  such that   $$ (x-a)(x-b)(x-c)(x-d)\geq 0 $$   for every balanced quadruple  $(a,b,c,d)$ .  
(Ivan Novak)

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