CN-SMO 2012 Problem 4

Let $a, b, c, d$ be real numbers satisfying inequality $a\cos x+b\cos 2x+c\cos 3x+d\cos 4x\le 1$ holds for arbitrary real number $x$ . Find the maximal value of $a+b-c+d$ and determine the values of $a,b,c,d$ when that maximum is attained.