Inequality of Numerators

Algebra Level 2

Let $a,b,c$ be positive reals. Also, let $k$ be the largest possible real such that

$\dfrac{a}{1}+\dfrac{b}{1}+\dfrac{c}{1}+\dfrac{a+b}{1}+\dfrac{b+c}{1}+\dfrac{c+a}{1}\le \dfrac{a+b+c}{k}.$

If $k$ can be expressed as $\frac{p}{q}$ for relatively prime positive integers $p$ and $q$, then what is $p+q?$

×