Let $x_1(t)$ and $x_2(t)$ be two low$-$pass signals bandlimited to $500$ Hz and $1000$ Hz respectively and let $($i$)y(t)=x_1(t)+x_2(t),($ii$)y(t)=x_1(t)x_2(t),$ and $($iii$)y(t)=x_1(t)**x_2(t).$ The signal $y(t)$ is sampled using ideal sampler at intervals of $T_s$ seconds. Determine the maximum sampling intervals $T_s$ such that $y(t)$ is recoverable from its sampled version $y_s(t)$ through the use of an ideal low$-$pass filter.