  Resetting is a strategy for boosting the speed of a target-searching process.
Since its introduction over a decade ago, most studies have been carried out
under the assumption that resetting takes place instantaneously. However, due
to its irreversible nature, resetting processes incur a thermodynamic cost,
which becomes infinite in case of instantaneous resetting. Here, we take into
consideration both the cost and first passage time (FPT) required for a
resetting process, in which the reset or return to the initial location is
implemented using a trapping potential over a finite but random time period. An
iterative generating function and counting functional method \`a la Feynman and
Kac are employed to calculate the FPT and average work for this process. From
these results, we obtain an explicit form of the time-cost trade-off relation,
which provides the lower bound of the mean FPT for a given work input when the
trapping potential is linear. This trade-off relation clearly shows that
instantaneous resetting is achievable only when an infinite amount of work is
provided. More surprisingly, the trade-off relation derived from the linear
potential seems to be valid for a wide range of trapping potentials. In
addition, we have also shown that the fixed-time or sharp resetting can further
enhance the trade-off relation compared to that of the stochastic resetting.
