If  $y=x+1$ is axis of parabola, $x+y=4$  is tangent of same parabola at its vertex and $y=2x+3$  is one of its tangents, then the length of the latus rectum of parabola is $\frac{a\sqrt{2}}{b}$ where a and b are relatively prime natural numbers, then $\frac{a+b+1}{12}=$