
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 \((a, b, c, d)\).

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