$1.$ Please design rules for the following Turing Machine: it takes a string of $1$s and $0$s as an input and keeps the first digit as it is$,$ but every other digit converts to Boolean complement. For instance, $1100$ will be converted to $1011,$ and $0001$ will be converted to $0110.$

$2.$ Please design rules for the following Turing Machine: it ends in $$reject$$ state after detecting $0$ in a string and bringing the cursor to the beginning of the tape, i$.$e$.$ to delta. However, if the input string has only $1$s this Turing Machine bring the cursor to the beginning of the tape, i$.$e$.$ to delta, and ends in the state $$accept$.$

$3.$ Please, explain a design idea for a TM that takes a binary number n ending with $0$ as an input and outputs n$+1.$In other words, this TM adds one to an input string viewed as a binary number. Examples If n$=100,$ then n$+1=101$ If n$=100$ then n$+1=101$