11 days. Just for clarification, does this include your arrival and departure days from France, but not your departure day from North America? If so, that would mean 10 nights in France, but only 9 full days in France.
I will assume you have 11 full days in France and 12 nights given your description of your travel. 6 days in Paris sounds good. Full days or does that include your arrival day? Arrival day will be a truncated day due to time from the airport, getting to your hotel and fatigue.
Agree with others that the Loire is too far as a day trip, so you should spend, perhaps two days and three nights there, at a minimum. See some chateaux, some countryside, nights in the city of Tours or smaller towns - many to choose from, not limited to Blois or Amboise; we also enjoyed Azay le Rideau and Chinon. Apart from the many chateaux in the area, there are wineries to visit and cave dwellings near Saumur, as well as the medieval section of Tours.
We spent 7 nights in the area, cycling from Blois to Saumur, visiting one winery, seven chateaux, Tours’ medieval center and Fontevraud Abbey where the remarkable Aleinor of Acquitane’s tomb lies next to her husband’s (Henry II of England) and her favorite son’s, Richard the Lionhearted.
So, if you have six nights in Paris that includes five full days in Paris plus a sixth (CORRECTION - 7th) night in Paris for your day trip to Versailles, plus three nights in the Loire, that would be 9 (CORRECTION 10) of your 10 or 12 nights. That 10th or 12th night would be in or near Paris for your flight home, which maybe leaves you just two nights for a brief trip to somewhere else.
If you have only 10 nights in France, you’re tapped out. If you do have just one other night, just add that night to Paris or the Loire, with the option of a day trip to Chartres, Fontenbleu, Giverny, or wherever is close and suits your fancy.
If you have 12 nights, then you might consider a short visit to just one of the areas that you mention.
One can never do it all, so slow it down and enjoy it more. Spend less time on one to two hour train journeys in each direction plus another one or two hours on busses or metro trains in Paris getting to and from the RER or TER trains that take you on “day trips.”
LATE NOTE - it may make sense to split up your Paris time. Say, 4 nights in Paris, then 3 in the Loire, then the balance back in Paris.