6.4. To be freed from Ivan the Fisherman's net, the Golden Fish gives him a reward: either 4 pearls and 1 diamond; or 2 emeralds and 2 diamonds; or 2 pearls, 1 emerald, and 1 diamond. After several releases of the Golden Fish, Ivan the Fisherman received 1000 pearls, 800 emeralds, and some number of diamonds. How many times did the Golden Fish get caught in Ivan the Fisherman's net?

Answer, option 1. 900.

Answer, option 2. 550.

Answer, option 1. 1400.

Answer, option 2. 650.
Solution 1. Let there be $x$ actions of the first type, $y$ actions of the second type, and $z$ actions of the third type. From the problem statement, we get two equations $4 x+2 z=2000, 2 y+z=800$, and we need to find $x+y+z$. By adding the first equation to twice the second, we get $4 x+4 y+4 z=3600$, from which $x+y+z=900$.

Remark. The situation described in the problem is indeed possible, for example, with $x=499, y=399$ and $z=2$. There are many other options.