$(0, 0)$, on an infinite cartesian plane. At time $t = 0$, it divides and replicates itself, moving exactly $1$ unit in each of the $8$ major compass directions (N, NE, E, SE, S, SW, W, and NW), leaving $(0, 0)$ vacant.

A single snowflake rests at the origin,At $t = 1$, all snowflakes which occupy the *exact* same points are merged into one, and then the process repeats for every snowflake, resulting in the following pattern:

So, to iterate the first few steps:

- There is $1$ snowflake at time $t = 0$
- At time $t = 1$, there are $8$ snowflakes pre-merge.
- No snowflakes coincide, so there are $8$ snowflakes post-merge as well.
- At time $t = 2$, there are $64$ snowflakes pre-merge.
- Post-merge, only $33$ snowflakes remain.

How many snowflakes are there at time $t = 1000$, post-merge?