A positive integer $x$ satisfies the following: \[\{\frac{x}{3}\}+\{\frac{x}{5}\}+\{\frac{x}{7}\}+\{\frac{x}{11}\}=\frac{248}{165}\]Find all possible values of \[\{\frac{2x}{3}\}+\{\frac{2x}{5}\}+\{\frac{2x}{7}\}+\{\frac{2x}{11}\}\]where $\{y\}$ denotes the fractional part of $y$.
Problem
Source: 2024 Israel National Olympiad (Gillis) P2
Tags: number theory, remainders, fractional part, national olympiad