Symmetric Matrices & Quadratic Forms · Superchallenge problem

Find the largest constant $c$ such that $x_1^2 + x_2^2 + x_3^2 + x_4^2 \ge c\,(x_1x_2 + x_2x_3 + x_3x_4)$ for all real $x_1, x_2, x_3, x_4$.