I want to find out the steps and intuition into how we come up with the Inverse Fourier transform. I can't just accept it as is, I want to find the complete derivation from the forward to the inverse. Most solutions I found was just proving that the inverse Fourier transform is true, but proving already assumes an existing solution.
Formally, from the forward Fourier transform which is in the form
$$f(n) = \int_{-\infty}^{\infty}f(t)e^{-2\pi nit}dt$$
How can we get the inverse which is
$$f(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty}f(n)e^{2\pi nit}dn$$
Note that I wish to see the complete thinking invlove, not the proof why this is true but the steps that resulted it.