I have this question:
$L = \{\langle M \rangle | M$ is TM that accepts every palindrome over its alphabet $\}$
Proof that $L$ is not Turing-recognizable by showing reduction from other non Turing-recognizable language.
What I have tried to do is to show reduction from $A_{TM}$ (Accept problem) to $\bar L$, and I didn't succeed to show one. I tried to create a new machine that will reject some palindrome(s) if $M$ accepted $w$ when $\langle M,w\rangle$ is the input of $A_{TM}$. But if $M$ will not halt on $w$, then the palindrome(s) will reject also by mistake.
So how can I show reduction from $A_{TM}$ to $\bar L$ (or $\bar {A_{TM}}$ to $L$)?