Polynomial's Common Multiple

Algebra Level 2

$\begin{aligned} f(x) &=& p(x)(x-1) +1 \\ f(x) &=& q(x)(x-2) +1 \\ f(x) &=& r(x)(x-3) +1 \end{aligned}$

Let $f(x)$ be a cubic monic polynomial, and $p(x), q(x), r(x)$ be quadratic polynomials satisfying the system of equations above.

What is the value of $f(4)?$