Answer: 14

Set up the common landing marks.

At 8 AM all three are on mark $199$. After that:

• $X$ lands on $199, 203, 207, \ldots$ (step $4$)
• $Y$ lands on $199, 205, 211, \ldots$ (step $6$)
• $Z$ lands on $199, 204, 209, \ldots$ (step $5$)

A mark $m$ is a common landing iff $m - 199$ is a multiple of $\text{lcm}(4, 6, 5) = 60$.

List common landings in the range $195 < m < 1000$.

$m = 199 + 60k$

$199,\ 259,\ 319,\ 379,\ 439,\ 499,\ 559,\ 619,\ 679,\ 739,\ 799,\ 859,\ 919,\ 979$

The next one, $1039$, exceeds $1000$.

Count: $\dfrac{979 - 199}{60} + 1 = 13 + 1 = 14$