ISL 1989 Problem 78

Let $P(x)$ be a polynomial with integer coefficients such that $$P(m_1) = P(m_2) = P(m_3) = P(m_4) = 7$$ for given distinct integers $m_1,m_2,m_3,$ and $m_4.$ Show that there is no integer m such that $P(m) = 14.$