CN-SMO 2021 Problem 5

To commemorate the $43rd$ anniversary of the restoration of mathematics competitions, a mathematics enthusiast arranges the first $2021$ integers $1,2,\dots,2021$ into a sequence $\{a_n\}$ in a certain order, so that the sum of any consecutive $43$ items in the sequence is a multiple of $43$ . (1) If the sequence of numbers is connected end to end into a circle, prove that the sum of any consecutive $43$ items on the circle is also a multiple of $43$ ; (2) Determine the number of sequences $\{a_n\}$ that meets the conditions of the question.