This paper considers a discrete time optimal stopping problem 
with a finite planning horizon. 
In addition to an offer that may appear randomly, at each point 
in time over the entire planning horizon a quitting offer is assumed 
to be available. By accepting this, a decision maker can terminate the process.
This paper assumes that the probability 
that an offer will appear if a search cost is paid is higher 
than if it is not paid. 
Thus, decisions must be made as to whether or not to accept the quitting offer, 
to accept an  appearing offer, and to conduct a search for an offer. 
The main purpose of this paper is to clarify the properties 
of the optimal decision rules. 
One of our main findings is that the quitting offer must either be accepted 
at the starting point of the process or not be accepted prior 
to the end of the planning horizon, the deadline.