OMM 2022 Problem 1

A number $x$ is "Tlahuica" if there exist prime numbers $p_1,\ p_2,\ \dots,\ p_k$ such that $$x=\frac{1}{p_1}+\frac{1}{p_2}+\dots+\frac{1}{p_k}.$$ Find the largest Tlahuica number $x$ such that $0<x<1$ and there exists a positive integer $m\leq 2022$ such that $mx$ is an integer.