The integers $-1$, $2$, $-3$, $4$, $-5$, $6$ are written on a blackboard. At each move, we erase two numbers $a$ and $b$, then we re-write $2a+b$ and $2b+a$. Find how many of the sextuples $(0,0,0,3,-9,9)$, $(0,1,1,3,6,-6)$, $(0,0,0,3,-6,9)$, $(0,1,1,-3,6,-9)$, $(0,0,2,5,5,6)$ can be gotten.

Let's think step by step and output the final answer within \boxed{}.
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