In Cartesian coordinates, two areas \(M\) and \(N\) are defined as follows:  
- \(M: y \geq 0, y \leq x, y \leq 2-x\);  
- \(N: t \leq x \leq t+1\).  
Here, \(t\) is a real number such that \(t \in [0,1]\).  

Determine the area of \(M \cap N\). The original answer is in the form \(-at^2 + bt + \frac{c}{d}\), where \(\frac{c}{d}\) is a simplified fraction. Please give the value of \(a + b + c + d\).

Let's think step by step and output the final answer within \boxed{}.
user