Determine one triple of positive integer Note that $\frac{1}{2}+\frac{1}{3}+\frac{1}{6}=1$ and $\frac{1}{6}-\frac{1}{7}=\frac{1}{6\times 7}$ and $\frac{1}{2}+\frac{1}{3}+\frac{1}{7}+\frac{1}{42}=1$. 
Determine one triple $(x,y,z)$ of positive integers with $1000<x<y<z<2000$ and $\frac{1}{2}+\frac{1}{3}+\frac{1}{7}+\frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{45}=1$.
A friend of mine asked me this question. I have only been able to guess one solution but $x,y,z$ doesn't fall into the range.
Note: This question is meant to be done within 10-15min and without a computer.

There are three solutions for
$\frac{1}{x}+\frac{1}{y}+\frac{1}{z} + \frac{1}{2}+\frac{1}{3}+\frac{1}{7}+\frac{1}{45} = 1$
in the required range, I
found them by exhaustive search:
$$x=1771,\;y=1932,\;z=1980$$
$$x=1806,\;y=1892,\;z=1980$$
$$x=1830,\;y=1891,\;z=1953$$
