AMC 12 2023b Problem 22

A real-valued function $f$ has the property that for all real numbers $a$ and $b,$$$f(a + b) + f(a - b) = 2f(a) f(b).$$Which one of the following cannot be the value of $f(1)?$

$\textbf{(A) } 0 \textbf{(B) } 1 \textbf{(C) } -1 \textbf{(D) } 2 \textbf{(E) } -2$