Multiple layered expression.

$f(x) = 3x + 4 \\ g(x) = 6x - 19$

$\begin{aligned} \implies f \circ g (x) &= 3 (6x-19) + 4 \\ &= 18x - 53 \\ \implies g \circ f \circ g (x) &= 6 (18x - 53) - 19 \\ &= 108x - 337 \\ \implies f \circ g \circ f \circ g (x) &= 3 (108x - 337) + 4 \\ &= 324x - 1007 \end{aligned}$

$\therefore \ f \circ g \circ f \circ g (a) = -35 \\ \implies 324a - 1007 = -35 \\ \implies 324a = 972 \\ \implies a = \boxed{3}$