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Let me forwarn you that your state of residence, California, is a very progressive state and advocates the idea that both parents have an equal right to custody; being a mother myself, I do not agree with this idea, but I do not live in California. However, the truth of the matter is that more an more states are deciding custody cases along this idea.
I do not think you should sign anything. If worse comes to worse, file a Complaint/Petition for Custody and ask for primary physical custody. State that the father will have liberal visitation rights but with his travelling, the child will not have the benefit of being with the father as he would suggest. Also state that you are prepared to care for your child 100% of the time and do not think it's a good idea for the child to be with a nanny - that defeats Mr. _____ entire argument for seeking custody.
Judges in all 50 States apply the same standard and the same test in awarding custody: They decide what they believe will be "in th best interests and permanent welfare of the child." So use that phrase a few times in your Complaint for Custody.
If it gets before a Judge because you have not been able to agree, have your lawyer question him on his work schedule, travel schedule, etc. If it comes down to having the child be with a nanny versus his mother, of course the Judge will pick the mother.
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Thank you for allowing me to assist you,
ANDREA, JD, LL.M., Taxation
Member, NY & PA Bar
DISCLAIMER: The facts stated herein are information only and are not intended, nor should be construed as legal advice or legal opinion and No Attorney-Client relationship is formed by asking questions and receiving information; laws vary from state to state and to protect your interests, you should seek local counsel