The polynomial \(ax^3 - 20x^2 + x + 3\), where \(a\) is a constant, is denoted by \(p(x)\). It is given that \((3x + 1)\) is a factor of \(p(x)\).

(i) Find the value of \(a\).

(ii) When \(a\) has this value, factorise \(p(x)\) completely.