3-pile Sharing Nim and the linear time winning strategy

We study a variant of 3-pile {\sc Nim} in which a move consists of taking tokens from one pile and, instead of removing, topping up on a smaller pile provided that the destination pile does not have more tokens then the removed pile after the move. We discover a situation in which finite sequences of Sprague-Grundy values are palindromes. We establish a formula for $\P$-positions by which winning moves can be computed in linear time. We prove a formula for positions whose Sprague-Grundy values are 1 and estimate the distribution of those positions whose nim-values are $g$. We discuss the periodicity of nim-sequences that seem to be bounded.