1’s For All

The complexity of an integer is the minimum number of $1$'s needed to represent it using only addition, multiplication and parentheses. For example, the complexity of $2$ is $2$ (writing $2$ as $1+1$) and the complexity of $12$ is $7$ (writing $12$ as $(1+1+1)\times (1+1+1+1)$). We'll modify this definition slightly to allow the concatenation operation as well. This operation (which we'll represent using ©) takes two integers and "glues" them together, so $12\$©$\ 34$ becomes the four digit number $1234$. Using this operation, the complexity of $12$ is now $3$ (writing it either as $(1 \$©$\ 1) + 1$ or $1\$©$\ (1+1)$).  Note that the concatenation operation ignores any initial zeroes in the second operand: $1\$©$\ 01$ does not result in $101$ but results in $11$.

We'll give you $1$ guess what the object of this problem is.

Each test case consists of a single line containing an integer $n$, where $0 < n \leq 100\\, 000$.

Output the complexity of the number, using the revised definition above.