  Combinatorics of Kazhdan-Lusztig cells in affine type $A$ was originally
developed by Lusztig, Shi, and Xi. Building on their work, Chmutov, Pylyavskyy,
and Yudovina introduced the affine matrix-ball construction (abbreviated AMBC)
which gives an analog of Robinson-Schensted correspondence for affine symmetric
groups. An alternative approach to Kazhdan-Lusztig theory in affine type $A$
was developed by Blasiak in his work on catabolism. He introduced sign
insertion algorithm and conjectured that if one fixes the two-sided cell, the
recording tableau of sign insertion process determines uniquely and is
determined uniquely by the left cell. In this paper we unite these two
approaches by proving Blasiak's conjecture. In the process, we show that
certain new operations we introduce called partial rotations connect the
elements in the intersection of a left cell and a right cell. Lastly, we
investigate the connection between Blasiak's sign insertion and the
standardization map acting on the set of semi-standard Young tableaux defined
by Lascoux and Sch\"utzenberger.
