Ward and Gabriëlle are playing a game on a large sheet of paper. At the beginning, there are 999 ones written on the paper. Ward and Gabriëlle take turns, with Ward starting. A player whose turn it is may choose two numbers $a$ and $b$ from the paper for which $\operatorname{ggd}(a, b)=1$, erase these numbers, and write the number $a+b$ on the paper. The first one who cannot make a move loses. Determine who of Ward and Gabriëlle can certainly win this game.
Gabriëlle can win with the following strategy: she always chooses the two largest numbers on the paper as $a$ and $b$. We prove by induction on $k$ that she can always do this and that after her $k$-th move, the number $2k+1$ and further $998-2k$ ones are on the paper.

In Ward's first turn, he can only choose $a=b=1$. After that, the number 2 and 997 ones are on the paper. Gabriëlle now chooses the two largest numbers, so $a=2$ and $b=1$, resulting in the number 3 and 996 ones. This proves the base case $k=1$. Suppose now that for some $m \geq 1$, after Gabriëlle's $m$-th turn, the number $2m+1$ and $998-2m$ ones are on the paper. If $998-2m=0$, then Ward cannot make a move. Otherwise, Ward has two options: choosing $a=b=1$ or $a=2m+1$ and $b=1$. We consider these two cases separately:

- If Ward chooses $a=b=1$, the situation becomes: the number $2m+1$, the number 2, and $996-2m$ ones. Gabriëlle now chooses the two largest numbers again, so $a=2m+1$ and $b=2$ (this is allowed because the gcd is 1). After her move, the number $2m+3=2(m+1)+1$ and $996-2m=998-2(m+1)$ ones remain.
- If Ward chooses $a=2m+1$ and $b=1$, the situation becomes: the number $2m+2$ and $997-2m$ ones. Gabriëlle now chooses the two largest numbers again, so $a=2m+2$ and $b=1$ (this is allowed because the gcd is 1). Note that there is still a 1 available, as $997-2m$ cannot be 0 because it is odd. After her move, the number $2m+3=2(m+1)+1$ and $996-2m=998-2(m+1)$ ones remain.

This completes the induction.
We see that Gabriëlle can always make a move. After Gabriëlle's 499th move, the paper contains only the number 999, so Ward cannot make a move and Gabriëlle wins.