**Category:** Algebra  
**Subtype:** Intersection of functions  
**Problem Description:** Find all values of \(m\) such that the line \(d: y = x + m - 1\) intersects the graph of the function \(y = \frac{2x + 1}{x + 1}\) at two distinct points \(M\) and \(N\) with a distance of \(MN = 2\sqrt{3}\).