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You need to be a permanent resident of any state for 6 months before you can file for divorce in that state. If you file for divorce in NM, the case will probably end up getting dismissed if the court determines that you still have permanent residences in Michigan. For international travel, you will need written permission of the other parent to get a passport for the child. For travel within the US, you should get the permission of the other parent before you take the child out of state -- simply because if one parent takes the child out of the home state and the other parent goes to court at that time and protests the travel, the court is most likely to order that the child be brought back into the state and/or custody will be turned over to the other parent at that time.
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