Hanoi Open Mathematics Competition 2013 Senior Question 9

A polynomial \(P(t)=t^3+at^2+bt+c\) has three distinct real roots. If \((x^2+x+2013)^3+a(x^2+x+2013)^2+b(x^2+x+2013)+c=0\) has no real roots, prove that \(P(2013)\gt 1/64\).