Construct a DFA in $8$ or fewer states, i$.$e$.$ I$$m fine if you delete unreachable states, that is the product construction of two DFAs $$ one that recognizes an odd number of $1$s and one that recognizes a binary number with a leading $0$ and ending $1.$

There is a corresponding test file p$2$test.txt:

$01$

$0111$

$0101001$

$001011$

$1$

$0101$

$11000110$

$0010$

$10101010$

First $4$ should accept, the second $5$ reject.