$(1)$ Give a string win${}_4^{**}$ satisfying ${}_2(11001110100)={}_4(w).$

$(2)$ Give a string win${}_2^{**}$ satisfying ${}_2(w)={}_4(130321).$

$(3)$ Draw the minimum DFA that accepts the language $\{\begin{array}{cc}\text{w}\text{i}\text{n}& {}_2^{**}|{}_2(w)\end{array}$ is a multiple of $3$