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The back of the Paris postcard reads: Miss Cassatt. (Girl in Garden) Bequest of Mr. P. 1937. Let's get to the museum to see what this could mean.

The Musee d'Orsay opens at 9:30am today and is just a 5-minute walk from the hotel. Admission is €14 ($15.25) and thus is beyond what you have left ... unless you earned some money busking.
However ... I have a sneaky idea. Tell the person at the ticket counter that you are a new member of the AFMO (American Friends Musee d'Orsay) at the $250 "friend" level. If they ask for membership info, suggest they call the New York office and ask for Christine who handled your membership. Since it's the middle of the night in New York, they would be unable to call and you should be off the hook.
Once you're in the museum, take a look at the Impressionist works by Miss Cassatt currently housed there. Click on the images to enlarge them enough to read the descriptions so you may locate the painting in question: