problem$1$: Use the general algorithm that we learned in class to design a non$-$deterministic finite automaton that recognizes the language $1(1$ U a$)*$ of all the words that start with $1$ followed by symbols $1$ and a$.$

$1$ and a are languages consisting of only one $1-$symbol word each: a is a language consisting of a single $1-$symbol word a; $1$ is a language consisting of a single $1-$symbol word $1,$ etc.;

for any two languages C and D$,$ the notation CD means concatenation;

transform the resulting non$-$deterministic finite automaton into a deterministic one.

problem $2$: Apply the general algorithm for transforming the finite automaton into a regular language $($i$.$e$.,$ a language described by a regular expression$)$ to Automaton B from Problem $1\mathrm{.}4.$ For simplicity, assume that we only have symbols $1,$ a$,$ and $?.$ Eliminate first the state n$,$ then the state w$.$

$***$PLEASE GIVE ANSWER WITH DRAWINGS$***$