Turing machine reduction task
Given a TM $T$, consider the machine $T'$ which, on input $w$, simulates $T$ on $w$ and enters a special state $q$ if and only if has determined that $T$ accepts $w$. Note you can easily guarantee $T'$ never enters $q$ prior to $T$ accepting $w$. Moreover, you can make sure that, once $T'$ enters $q$, it starts to cycle between all its states indefinitely. (For instance, have $T'$ write a special tape symbol so it knows it is supposed to do so.)

For an arbitrary $T$, producing the description of $T'$ is computable. I'll leave it to you to fill in the gaps so this is a full-fledged reduction.



As a rule of thumb, these exercises can usually be solved by producing description of machines (in this case, $T'$) which encode the answer to the original problem ($A_{\mathrm{TM}}$) in their own behavior. In this particular case, the idea is to establish the equivalence between $T'$ entering all its states and $T$ accepting $w$.