You are trying to build your query based on the values that those $a_* variables have. But you are passing them inside single quotes, and what goes inside single quotes is not interpolated.

No, only the outermost quotes determine whether interpolation will happen or not. Since those are double quotes it will work fine. Minimal example:

my $var = test';
my $result = "This is a '$var' again";
print "$result\n";
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$var will be interpolated and This is a 'test' again will be printed.