Example 6 Let $a, b, c$ be three distinct real numbers, and $p(x)$ be a polynomial with real coefficients. It is known that: (1) $p(x)$ divided by $x-a$ yields a remainder of $a$; (2) $P(x)$ divided by $x-b$ yields a remainder of $b$; (3) $p(x)$ divided by $x-c$ yields a remainder of $c$. Find the remainder when the polynomial $p(x)$ is divided by $(x-a)(x-b)(x-c)$.
(1990 Italian Competition Problem)