CN-TST 2012 Problem 15

Find the smallest possible value of a real number $c$ such that for any $2012$ -degree monic polynomial $$P(x)=x^{2012}+a_{2011}x^{2011}+\ldots+a_1x+a_0$$ with real coefficients, we can obtain a new polynomial $Q(x)$ by multiplying some of its coefficients by $-1$ such that every root $z$ of $Q(x)$ satisfies the inequality $$\left\lvert \operatorname{Im} z \right\rvert \le c \left\lvert \operatorname{Re} z \right\rvert.$$