Consider the following languages:

EONE$\_$TM $=\{|$ M is a TM and M accepts exactly one string that starts with $1\}$

SONE$\_$TM $=\{|$M is a TM and M accepts exactly some strings that start with $1\}$

AONE$\_$TM $=\{|$ M is a TM and M accepts exactly all strings that start with $1\}$

A$\_$TM $=\{|$ M is a TM and accepts string w$\}$

$(1)$Using these languages, construct a TM F that is a reduction from A$\_$TM to SONE$\_$TM

$(2)$Consider the language SONE $\_$TM and AONE$\_$TM from previous question and consider the following TM F:

F $="$On input where M is a TM:

$1.$ Output

$$Is the above TM F a reduction from AONE$\_$TM to SONE$\_$TM$?($If you think it is$,$ prove it$.$ If you think it is not, give a counter example$)$