I $\heartsuit$ You!... well, almost

Calculus Level 3

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 $a + b + c.$

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