# Brilli's triangular search

Brilli the ant is on a search for food to bring back to his anthill. He starts his search from a point one space above a lattice in the shape of an equilateral triangle with side length 96. A similar lattice is shown in the picture, but this lattice has side length 4.

He searches the lattice by checking each point for food, and then moving to the next point on the lattice. He moves in a zig-zag pattern as shown in the picture. Once he finds food, he goes back to the starting point along the shortest lattice path that he can find.

If there is exactly one point in the triangular lattice that contains food, and each point has equal chance of containing food, then what is the expected value of the distance of Brilli's journey?

Notes and clarifications: The starting point will not contain food. When Brilli moves back to the starting point, he moves along the lattice points (not in a straight line to the starting point), but he does not have to retrace the path he used to get to the food.

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