OMM 2000 Problem 3

Given a set $A$ of positive integers, the set $A'$ is composed from the elements of $A$ and all positive integers that can be obtained in the following way: Write down some elements of $A$ one after another without repeating, write a sign $+$ or $-$ before each of them, and evaluate the obtained expression. The result is included in $A'$ . For example, if $A = \{2,8,13,20\}$ , numbers $8$ and $14 = 20-2+8$ are elements of $A'$ . Set $A''$ is constructed from $A'$ in the same manner. Find the smallest possible number of elements of $A$ , if $A''$ contains all the integers from $1$ to $40$ .