MathNet 2012 Question 9

Given that $P$ is a real polynomial of degree at most $2012$ such that $P(n) = 2^{n}$ for $n = 1, 2, \ldots, 2012$, what choice(s) of $P(0)$ produce the minimal possible value of $P(0)^{2} + P(2013)^{2}$?