Alice and Basilio make \(20\) bills. On each bill they will write a seven-digit number, so each bill initially has \(7\) empty cells for digits.
Basilio repeatedly calls out a digit, either \(1\) or \(2\). After each call, Alice writes this digit into any empty cell of any bill and shows the result to Basilio. When all cells are filled, Basilio takes as many bills with different numbers as possible (if several bills have the same number, he takes only one of them), and Alice keeps the rest.
What is the largest number of bills Basilio can guarantee to get, no matter how Alice plays?