Write a Polynomial in Terms of Its Roots

(2015 AIME I #10) Let $f(x)$ be a third-degree polynomial with real coefficients satisfying $$\left|f(1)\right| = \left|f(2)\right| = \left|f(3)\right| = \left|f(5)\right| = \left|f(6)\right| = \left|f(7)\right| = 12.$$ Find $\left|f(0)\right|.$