On one problem of the optimal choice of record values

Independent random variables $X_1,X_2,\dots, X_n$ having $\mathrm U([0,1])$-uniform distribution and the upper record values in this set are considered. We study the problem how to maximize (taking into account some consecutively observed values $x_1,x_2,\dots,x_k$ of these $X$-s) the expectation of sums of records in this sequence under the optimal choice of the corresponding variable $X_k$ (instead of $X_1$) as the initial record value.