Nemotron-IMO-Bench 2026 A8

Find the smallest real number $C>1$ with the following property: for any integer $n>1$ and any non-integers $a_1,...,a_n>0$ with $\sum_{i=1}^n 1/a_i=1$ there are positive integers $b_i\in \{\lfloor a_i\rfloor, \lfloor a_i\rfloor+1\}$ such that $1<\sum_{i=1}^n 1/b_i\leq C$.