It is possible your mother can give you a power of attorney. You would want an need a "durable" power of attorney for health care and to care for her estate. If you do an internet search for California Durable Power of Attorney you should find some good forms with explanations.
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If she is too far gone to help herself, you might have to get the Court to appoint you as her Conservator. That makes you responsible for her, just like a child, and she cannot make any more legal decisions. If you waste her money, you can be sued by your sister or the Court. Measure twice cut once.
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If she is too far gone to make decisions, then she can not sign a durable medical or durable financial power of attorney and you must seek to have a guardianship and a conservatorship set up.
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