Say the left-most letter of the string is on place $1$ and the right-most letter is on place $4044$. The other places are $2$ to $4043$ respectively from left to right. Bilbo tells Gandalf, that he will use $c$ as a separator and will put them on places $2k, 1\leq k \leq 2022$. Moreover, he will use $a$ to indicate $1$ in binary representation and $b$ to indicate $0$ in binary representation. These will be put on places $2k+1, 0\leq k \leq 2021$. Bilbo also tells Gandalf that there is one exception: if the number Nitzan chooses is $2^{2022}$, then his string will only consist of $a$'s. Gandalf will thus immediately know Nitzan's number, if he gets a empty string or a string with only $a$. When Bilbo sees Nitzan's number and it is not $2^{2022}$, he converts it to it's binary representation, then to it's $a,b$-representation and puts it's last letter (left-most letter) on place $1$, the letter besides it on place $3$ and so on (note that numbers $1$ to $2^{2022}-1$ need at most $2022$ digits in binary, hence we have enough places to represent these numbers). Now, if Nitzan decides to remove all $c$s, then Gandalf sees a string in $a,b$-representation, which he converts to binary and then to decimal and he knows Nitzan's number. If Nitzan removes $a$ or $b$, then Gandalf sees, due to the separators $c$, where these $a$s or $b$s are missing. He thus puts them back, converts the number in $a,b$-representation to binary and then to decimal and again knows Nitzan's number. Hence we conclude, that Bilbo and Gandalf can work together s.t. Gandalf always knows Nitzan's number, no matter how he acts. $\square$