Modular Madness

Let $a$ and $b$ be positive integers such that $a^2+b^2 \equiv 2014 \pmod{a+b}$ and $a^3 + b^3\equiv 2014^2 \pmod{a^2+b^2}$ . Evaluate the digit sum of

$\dfrac{a^3+b^3}{a+b}$ .