A reverse split does not increase the number of shares available to print.
The number of authorized shares must shrink at the same ratio of the reverse split. If they do not, as in the first reverse split, a shareholder vote must occur because that changes the capital structure of the company.
In the first reverse split, naive shareholders thought they were voting on the split itself. They weren't since that doesn't require a shareholder vote. They were actually voting on changing the capital structure by leaving the number of authorized shares intact.
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