NSolve and NIntegrate, or a better approach

I need to define and plot the following function

$ $ a(t) := \exp\left(\int Z(t)\; dt\right) $ $

where $ Z(t)$ is the solution to the equation

$ $ 0 = t – 2 \int^Z_1 F(x)\; dx $ $

with $ F = F(x)$ being a known (but complicated and non-integrable) function.

How do I define and plot the function $ a(t)$ in Mathematica?

Here is my attempt with a particular function $ F(x)$ that I need to work with:

A = 0; F[x_] = - ((4*A*x^(9/2) + 64*x^6 - 4*Sqrt[A*x^9*(32*x^(3/2) + A)])^(1/3)/(16*x^4 - 4*x^2*(4*A*x^(9/2) + 64*x^6 - 4*Sqrt[A*x^9*(32*x^(3/2) + A)])^(1/3) - (4*A*x^(9/2) + 64*x^6 - 4*Sqrt[A*x^9*(32*x^(3/2) + A)])^(2/3)));  A = 0; Int[Z_?NumericQ] := NIntegrate[F[x], {x, 1, Z}] S[t_?NumericQ] := NSolve[t - 2*Int[Z] == 0, Z] a[t_] := Exp[Integrate[S[t], t]] 

However, when trying to evaluate for example $ a(2)$ I get the following error:

The error