Triangular Pick's Theorem

Pick’s theorem states that, given a polygon with vertices on a unit square lattice, the relationship between the area \(A_S\) of the polygon, the number of boundary points \(B\) of the polygon, and the number of interior points \(I\) of the polygon is \(A_S = I + \frac{1}{2}B - 1\).

There exists a similar relationship for a polygon with vertices on an equilateral triangle lattice in which the area of the smallest possible triangle is 1.

The area of the shaded triangle is 1.

If \(A_T\) is the area of the polygon, \(B\) is the number of boundary points of the polygon, and \(I\) is the number of interior points of the polygon, then the relationship is \(A_T = aI + bB + c.\)

What is \(|a|+|b|+|c|?\)