Let us consider a polynomial $P(x)$ with integer coefficients satisfying:
\[
P(-1) = -4, \\
P(-3) = -40, \\
P(-5) = -156.
\]
What is the largest possible number of integers $x$ satisfying:
\[
P(P(x)) = x^2?
\]

Let's think step by step and output the final answer within \boxed{}.
user