$(60$ points$)$ Let $M1$ and $M2$ be the DFAs pictured below, accepting languages L $1$ and L $2,$

respectively. Draw DFAs accepting the following two languages

a$.$ L$1$ U L$2$

b$.L1L2$

$($b$)$

$(40$ points$)$ Draw a DFA accepting $\{ab,aa{\}}^{*}\{b{\}}^{*}\{a{\}}^{+}$