Recognizable Languages

Consider the following languages

A$^\{\backslash$forall$\}=\backslash$left$\backslash \{|$M$\backslash \backslash$text$\{$is a TM that accepts all strings of length less than k$\}\backslash$right$\backslash \}$

and

A$^\{\backslash$exists$\}=\backslash$left$\backslash \{|$M$\backslash$text$\{$ is a TM that accepts some strings of length less than k$\}\backslash$right$\backslash \}$

Note the difference between the languages all strings vs some strings

$1)$We want to show that A$^\{\backslash$exists$\}$ is recognizable.

We have shown that A$^\{\backslash$forall$\}$ is recognizable $($you can find the solution under the final exam study guide $).$

Explain why the TM used to show A$^\{\backslash$forall$\}$

is recognizable does not work for A$^\{\backslash$exists$\}$

$?$ How do you modify it to get a TM to recognize A$^\{\backslash$exists$\}.$

$.$