Design a Pushdown Automaton $($PDA$)$ recognizing each of the following languages and draw its state diagram.

Note that the transition function must be in the format of $"a,bc",$ namely we can only push$/$pop one symbol into$/$from the stack one time upon one input symbol. You will receive $0$ point if you don't following this instruction.

$(1)\{w|w=a^{n+m}{b}^{2n}{c}^{3m},n1,m0\}$

$(2)|=a^i{b}^j{c}^k$ or $\{$:$k=3j,i,j,k0\}$