  We study the efficiency of a two-channel reaction between two walkers on a
finite one-dimensional periodic lattice. The walkers perform a combination of
synchronous and asynchronous jumps on the lattice and react instantaneously
when they meet at the same site (first channel) or upon position exchange
(second channel). We develop a method based on a conditional first-passage
problem to obtain exact results for the mean number of time steps needed for
the reaction to take place as well as for higher order moments. Previous
results obtained in the framework of a difference equation approach are fully
confirmed, including the existence of a parity effect. For even lattices the
maximum efficiency corresponds to a mixture of synchronous events and a small
amount of asynchronous events, while for odd lattices the reaction time is
minimized by a purely synchronous process. We provide an intuitive explanation
for this behavior. In addition, we give explicit expressions for the variance
of the reaction time. The latter displays a similar even-odd behavior,
suggesting that the parity effect extends to higher order moments.
