Let
\[
I(x) = \int \frac{3\,dx}{(4x+6)\sqrt{4x^2 + 8x + 3}}
\]
and
\[
I(0) = \frac{\sqrt{3}}{4} + 20.
\]
If
\[
I\left(\frac{1}{2}\right) = \frac{a\sqrt{2}}{b} + c,
\]
where \(a, b, c \in \mathbb{N}\) and \(\gcd(a,b)=1\), then find the value of
\[
a + b + c.
\]