Given any collection of $2010$ nondegenerate triangles, their sides are painted so that each triangle has one red side, one blue side, and one white side. For each color, arrange the side lengths in order: let $b_1\le b_2\le\cdots\le b_{2011}$ denote the lengths of the blue sides; let $r_1\le r_2\le\cdots\le r_{2011}$ denote the lengths of the red sides; and let $w_1\le w_2\le\cdots\le w_{2011}$ denote the lengths of the white sides. Find the largest integer $k$ for which there necessarily exists at least $k$ indices $j$ such that $b_j$, $r_j$, $w_j$ are the side lengths of a nondegenerate triangle.