**Category:** Algebra  
**Subtype:** Polynomial Functions  
**Problem Description:** The problem involves finding the integer values of the parameter \(m\) such that there are exactly three tangent lines to the curve defined by the function \(y = x^3 + 3x^2 + 1\) that pass through the point \(A(1, m)\). The task is to determine the number of such integer values of \(m\).