$(10$ pts$)$ Assume that $H$ solves the Halting Problem for any arbitrary algorithm $M$ and input $w--$ i$.$e$.,$

if $M(w)$ halts, $H$ outputs 'yes';

if $M(w)$ doesn't halt, $H$ outputs $\text{'}$no$.\text{'}$

We now modify $H$ into a machine $H^{\text{'}}$ :

$($a$)$ put a "Xerox" machine on H$\text{'}$s front$-$end to convert the machine into a $1-$input

machine rather than a $2-$input machine;

$($b$)$ wherever $H$ has an instruction "output$(\text{'}$yes$\text{'})"$ replace that instruction with

$\{n$:$=1$

while $n>0$

$n$:$=1\}$

Now, here is the question you are to answer $($read it very carefully; it may not be asking what you

immediately think it's asking!$)$:

What is the behavior of $H^{\text{'}}$ on the input $x($or$,$ technically, $x\text{'}$s encoding: $)?$