Going around in circles

Question

Solve the grid. Then solve the grid again.

The answer can be one or two letters long.

Answer 1

Building off of @dvx2718’s answer below,

I started by trying to connect the sequence they got:
41965342862315732487264947952916
to digits of pi. I noticed that 1573 and 49479 are both backwards excerpts from pi (within the first hundred digits) but couldn’t find any other matches.

At that point, I remembered Jujustum’s hint about the solve path. I started by looking at the solution one row at a time, and realized that the filled-in numbers were placed in the same positions as the same digits in the decimal portion of pi. Below is the grid filled in with the missing digits, though it should be noted that there is an error in the penultimate cell.

Thus, as per dvx2718, the solution would seem to be

π / pi.

Answer 2

Speculative answer

I firmly believe this is the answer but I cannot prove it because a key piece is missing, yet everything else lines up. If I shouldn't be posting a speculative answer like this please let me know as I'm new to the site.

The answer is:

In one letter: π, or in two letters: pi

The reasoning:

First, I solved the sudoku.

Then, I note that there are exactly 32 cells to fill to solve this sudoku, and the word circle in the title makes me think of pi. Checking the first 32 digits of pi, we get 3.1415926535897932384626433832795, and the 33rd digit is 0. Since 0 cannot show up in sudoku, we see that from the beginning of pi, exactly 32 digits are usable in sudoku before we reach the first 0. Also, pi can be written using both a single letter π or two letters pi.

Here's the problem though:

I cannot find the relation between the solution to the sudoku and pi, other than the 32 digits thing. If we take the digits of the solution to the sudoku, that's 41965342862315732487264947952916, compare that with 3.1415926535897932384626433832795, the digits do not match.