Your current data structure makes this search harder than it needs to be.
Simple, easy-to-read code to work with this:

my @a2=map { $array[$_][0] } 0..$#array;
my ($goat_idx) = grep{ $a2[$_] eq "goat" } 0..$#a2;
print "Goat index is $goat_idx.\n";
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(Output is : Goat index is 2.) It will be "undef" if "goat" is not found.