Design a Turing Machine $($TM$)$ recognizing each of the following languages and draw its state diagram.

Note that each transition function of TM must be in the format of $"ab,\frac{L}{R}",$ namely the TM reads single symbol a from one cell of the tape, writes single symbol $b$ to the cell to replace $a,$ and then goes to either left $L$ or right $R.$

$(1)\{w|w=a^{3n}{b}^{2n},n0\}$

$(2)\{w|w=a^i{b}^j,ij,i,j>0\}$