  We introduce a cyclic proof system for the two-way alternation-free modal
$\mu$-calculus. The system manipulates one-sided Gentzen sequents and locally
deals with the backwards modalities by allowing analytic applications of the
cut rule. The global effect of backwards modalities on traces is handled by
making the semantics relative to a specific strategy of the opponent in the
evaluation game. This allows us to augment sequents by so-called trace atoms,
describing traces that the proponent can construct against the opponent's
strategy. The idea for trace atoms comes from Vardi's reduction of alternating
two-way automata to deterministic one-way automata. Using the multi-focus
annotations introduced earlier by Marti and Venema, we turn this trace-based
system into a path-based system. We prove that our system is sound for all
sequents and complete for sequents not containing trace atoms.
