
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 the smallest positive real number \( k \) that satisfies the following condition: For any four distinct real numbers \( a, b, c, \text{ and } d \) not less than \( k \), there exists a permutation \( p, q, r, \text{ and } s \) of \( a, b, c, \text{ and } d \) such that the equation
\[
\left(x^{2} + p x + q\right)\left(x^{2} + r x + s\right) = 0
\]
has exactly four distinct real roots.

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