Question $3.$ Prove that a language $A$ is Turing$-$recognizable if and only if $A{?}_m*A_{TM}-$

Structuring your proof: Proving the two implications separately is a good strategy.

$)>($ Assume $A$ is recognizable $($i$.$e$.,$ there's a TM whose language is $A),$ and provide a mapping reduction.

$(lArr)$ Assume $A{?}_m{A}_{TM}($i$.$e$.,$ there's a computable $f$ with winA iff $f(w)in{A}_{TM}),$ and provide a TM for $A$ to show $A$ is recognizable.