An algebra problem by A Former Brilliant Member

Algebra Level 4

$\large \prod_{n=1}^\infty (3^n)^{1/3^n} =3^{1/3} \times 9^{1/9} \times 27^{1/27} \times \cdots$

The product above can be expressed as $a^{b/c}$ , where $a$ is a prime number, and $b,c$ are coprime positive integers. Find $a + b +c$ .