Delay differential equation using fourier transform

I want to solve:

$$y'(t)+y(t)+y(t-1)=\delta (t)$$

The delta on the right hand side is used instead of the initial condition of $y(0)=1$

I then take the fourier transform of both sides:

$$Y(i w+1+e^{- iw})=1$$

Taking the inverse fourier transform of this expression is hopeless, I attempt to do it numerically. This is what $Y$ looks like (blue is real, red is imaginary) : This is its numerical inverse transform: (Don't worry about the oscillation near the discontinuities) But this clearly is not correct since the solution is supposed to look like this: I would like to learn what I have done wrong, how to do it correctly, and which problem I have actually solved (what is the function in the second image a solution to)

EDIT: I have used an optimization algorithm to estimate the function that represents the fourier transform of the actual solution and it looks like this:

$$\frac{0.431}{i2.704w+0.432+0.430e^{-i6.291w}}$$

$$\frac{0.431}{i2.704w+0.432+0.430e^{-i6.291w}}=\frac{1}{i2\pi w+1+e^{-i2\pi w}}$$
The $w$ was treated as cyclic frequency, which caused the discrepancy.