Maximum value 1

Algebra Level 4

$\large { a }^{ 4 }+{ b }^{ 4 }+{ c }^{ 4 }+abc\left( a+b+c \right) \ge k{ \left( ab+bc+ca \right) }^{ 2 }$

Find the maximum value of $k$ such that the inequality above holds true with all real values of $a,$ $b,$ and $c.$

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