PDA Design

$(50$ marks: half each$)$

Draw state diagrams for PDAs accepting the following languages $($provide brief explanations of how your machines work$)$:

$($a$)$

$L=\{a^i{b}^j{c}^k|i+k=j\}$

$($b$)$

$L=\{a^n(bc{)}^{2m}{a}^{n+m}|n,m0\}$