Determine constant $A$ such that $x[n] = x[n] star x[n]$

Let $x[n] = Adelta[n] – frac{sin(frac{3n}{2})}{pi n}$. Determine constant $A$ such that for all $n$ $$x[n] = x[n] star x[n] tag{1}$$

I think it’s not possible since $(1)$ leads to $$X(e^{jomega}) = X(e^{jomega})X(e^{jomega})$$ And this means $X(e^{jomega}) = 1$ or $X(e^{jomega}) = 0$. Also $$X(e^{jomega}) = begin{cases} A – 1 &0le | omega| le frac{3}{2} \ A & frac{3}{2}lt | omega| le pi end{cases}$$
It means no value of $A$ works. I don’t know whether is my answer correct. Maybe I’ve neglected something.