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To connect remotely, first determine the URL of the API for your installation. The default path is http://yourdomain.com/installpath/api/. Where yourdomain.com is the domain you've installed Blesta in, and installpath is the path to Blesta. If your server does not support .htaccess then the URL should instead appear as http://yourdomain.com/installpath/index.php/api/.
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