A tilting level was used to run precise levels to a construction project benchmark from benchmark DAM, which had an elevation of $345\mathrm{.}901$ m$.$ The line was run along a road that goes up a steep incline.

$$To expedite the job, backsight distances were kept to $50$ m while foresight distances were $25$ m$.$ The total length of differential levels was $5248$ m$.$ The measured elevation difference was $21\mathrm{.}051$ m$.$ What is the corrected project benchmark$$s elevation if the instrument has a sight line that declines at the rate of $0\mathrm{.}5$ mm per $50$ m$?$

$[8$ pts$]$