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use List::Util 'min';
use POSIX 'ceil';
...
use Memoize;
memoize 'num_ways';
print num_ways(100, 10, 667), $/;

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sub num_ways {
    my ($N, $S, $T, $callback, @so_far) = @_;
...
}

num_ways( 100, 10, 667, sub { print "@_\n" } );