I \(\heartsuit\) You!... well, almost

For Mother's Day, Little Albert wants to show just how much he loves his mom by making a card for her with a big, colorful heart on the front. He gets out his paper, uncaps a marker, and gets to work.

The curve that Little Albert drew is called a limaçon. (The extra loop inside the heart will be invisible to his mom after coloring.) The polar equation for the particular limaçon that he drew is \[r = 3 - 6 \sin \theta.\] The area of the region that Little Albert colored in is equal to \(a \pi + \frac{b \sqrt{3}}{c},\) where \(a,\) \(b,\) and \(c\) are positive integers, and \(b\) and \(c\) are coprime.

Find \(a + b + c.\)