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To be able to stay and get your green card without leaving, you will need to file an adjustment package. This includes the following forms: I-130 (Petition for family member), I-485 (application for Lawful Permanent Residency), I-765 (Application for work permit), I-131 (Application for Advance Parole), G-325a (Biographic data - one for each of you), I-693 (Medical exam that a certified doctor must fill out), and I-864 (Affidavit of support). You will need to file each form with supporting evidence and appropriate filing fees. You can find these forms at www.uscis.gov/forms.
The supporting evidence that you would file would be birth certificates, marriage certificate, divorce certificates if either of you have been married previously, proof of your spouse's U.S. Citizenship, proof of your legal entry into the U.S., and financial documents of your spouse to prove their income over the last year at least.
In about 3 to 5 months after filing, you should get a work permit and Advance Parole. About 5 to 7 months after filing you should get a marriage interview (though now it is taking longer, so maybe closer to 7 to 9 months). If all goes well, a few weeks later you should get a Residency status (green card).
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