Initially, there are $3^{80}$ particles at the origin $(0,0)$. At each step, the particles are moved to points above the $x$-axis as follows: if there are $n$ particles at any point $(x,y)$, then $\left\lfloor \frac{n}{3} \right\rfloor$ of them are moved to $(x+1,y+1)$, $\left\lfloor \frac{n}{3} \right\rfloor$ are moved to $(x,y+1)$, and the remaining to $(x-1,y+1)$. For example, after the first step, there are $3^{79}$ particles each at $(1,1)$, $(0,1)$, and $(-1,1)$. After the second step, there are $3^{78}$ particles each at $(-2,2)$ and $(2,2)$, $2 \times 3^{78}$ particles each at $(-1,2)$ and $(1,2)$, and $3^{79}$ particles at $(0,2)$. After $80$ steps, the number of particles at $(79,80)$ is:

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