The numbers \(1,2,3,4,5,6,7,8,9\) and \(10\) are written on a board.
In one move, you may erase any three of the remaining numbers, call them \(a,b\) and \(c\), and replace them with the three numbers \(2a+b,\; 2b+c\) and \(2c+a\).
Is it possible, after a sequence of such moves, for all \(10\) numbers on the board to be equal?