Maximize a Constant in an Inequality with a Restriction

Algebra Level 4

Real numbers $a$ , $b$ , and $c$ satisfy $abc=1$ . Given that $k$ is an integer $\geq 2$ , find the maximum possible $R$ that satisfies

$\dfrac{a^k}{a+b}+\dfrac{b^k}{b+c}+\dfrac{c^k}{c+a} \geq R$

for all $a$ , $b$ , and $c$ satisfying the condition previously stated.