Every planar graph is 1-defective (9,2)-paintable

Assume G is a graph, and k,d,m are natural numbers. The d-defective (k,m)-painting game on G is played by two players: Lister and Painter. Initially, each vertex has k tokens and is uncolored. In each round, Lister chooses a set M of vertices and removes one token from each chosen vertex. Painter colors a subset X of M which induces a subgraph G[X] of maximum degree at most d. A vertex v is fully colored if v has received m colors. Lister wins if at the end of some round, there is a vertex with no more tokens left and is not fully colored. Otherwise, at some round, all vertices are fully colored and Painter wins. We say G is d-defective (k,m)-paintable if Painter has a winning strategy in this game. We prove that every planar graph is 1-defective (9,2)-paintable. In addition, our winning strategy also implies that every planar graph is 1-defective [Formula presented]-paintable for any positive m.