What passport do you hold?
Right. I'm sorry to say that you would not be successful if you applied for a visa as a dependent of your cousin. Dependents visas are only available for parents or grandparents of those who have an immigration status and, in any event, must be completely or mainly financially dependent on their children/grandchildren.
As an American you will receive a visitor's visa on port of entry (ie. at customs) for 6 months but you must leave after that.
Sorry it could not be better news.
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