Find the length of the impulse response for the given output and input
You correctly figured out that the system must have $3$ zeros. So you just have to define a corresponding transfer function $H(z)$ that satisfies
$$H(z_i)=0,\qquad z_i=e^{j\omega_i},\quad i\in\{0,1,2\}$$
In other words, what is the minimum number of coefficients of a polynomial with $3$ zeros? Note that you might need to provide two solutions: one general solution, and one with the restriction that the polynomial coefficients be real-valued (as is required for a real-valued system).