Kristýna chose a certain odd natural number divisible by three. Jakub and David then examined triangles whose perimeter in millimeters was equal to the number chosen by Kristýna and whose sides had lengths in millimeters expressed by different whole numbers.

Jakub found such a triangle in which the longest side had the greatest possible length, and he wrote this value on the board. David found such a triangle in which the shortest side had the greatest possible length, and he also wrote this value on the board. Kristýna correctly added both lengths on the board, and it came to $1681 \mathrm{~mm}$.

Determine which number Kristýna chose.

(L. Růžičková)

Hint. Can you, given the described property of Kristýna's number, express Jakub's and David's numbers?

Translate the above text into English, retain the line breaks and formatting of the source text, and output the translation result directly.
Kristýna's number, denoted as $k$, is odd and divisible by three. This means that $k$ is an odd multiple of three, i.e.,

$$
k=3(2 n+1)=6 n+3
$$

for some natural number $n$. The side lengths (vm) of Jakub's and David's triangles correspond to triples of natural numbers that sum to $k$ and satisfy the triangle inequalities (the sum of any two numbers is greater than the third).

The largest possible length of the longest side in a triangle with perimeter $k$ corresponds to the largest natural number less than half of $k$ (due to the triangle inequality). Therefore, Jakub's number was $3 n+1$. The remaining two sides could correspond to, for example, $3 n$ and 2, but this does not concern us for solving the problem.

The largest possible length of the shortest side in a triangle with perimeter $k$ corresponds to the largest natural number less than one-third of $k$ (the corresponding triangle should be as close to equilateral as possible, but the sides must be different). Therefore, David's number was $2 n$; the remaining two sides corresponded to $2 n+1$ and $2 n+2$.

The sum of Jakub's and David's numbers is $5 n+1$, which, according to the problem statement, equals 1681. Thus, $5 n=1680$, and therefore $n=336$. Kristýna's number was

$$
k=6 \cdot 336+3=2019 .
$$