I invest for my real retirement, but regularly buy $5 a couple times a month to hit the PB, Mega, and State lotto. It's worth $10/mo to dream, and I don't check them right away, just next time I remember to bring them to the grocery store, so I get two weeks of fun and escapist thinking out of $5.
If the cash value was 360M after tax and taking it up front, there are a few things I'd do immediately, and a few things I'd think on. Besides doing all the safety and security stuff to protect my family, I'd probably have to admit to a small windfall to cover some of what I'd want to do.
I live in a really well off community (there are billionaires in my town) so having hundreds of millions would be unusual, but not unique. Where it would be a problem is my family is all in a very modest town, so I would help them in a way that didn't ruin everything.
I'd probably have to lie to my parents and tell them that I had a $2M ish real estate windfall to cover getting them better houses (not palaces) and paying them off. I really couldn't count on my mom not to run her mouth, which would cause more problems for her than for me, but she could say "my city son did well on a spec house and paid mine off" without putting herself in danger or attracting bad people to her, but if she slipped that i"m worth a few hundred million, it would cause a bunch of issues.
As for me, I would definitely quit our jobs immediately, we're already trying to find ways to have more time with our kids, and I'd simplify some stuff (sell off a rental house I have, or more likely, have it demolished and sell the lot so there's not liability).
I'd invest in timberlands/etc and for my own security, ensure I had a couple of self sustaining places that were unassailable even if I'm sued/etc. ie, set them up in trusts so I can't possibly have them taken away from me and could always "fail back" to them. This would make me sleep a lot better.
I would invest in self care, ensuring I had a workout/nutrition regimen and coaching and I would likely get an individual tutor for the kids to ensure they got the right education for what they're going to have to do. Again, I'm fortunate in that I'm in a community where I could get to some families with similar wealth and learn that whole world, even though I'm just a poor mill town boy done good.
Honestly, the key is setting it up so it doens't ruin you and also doesn't become a consuming thing. Probably invest the first 6 months in getting everything set up (family office/etc) and then focus more on living.
Yeah, and I'd probably finally get a plane, though instead of a traveler, I'd be way more likely to get a small, slow, fun plane for me, and have a private/charter for getting me places when i want to. I'd be pretty much done with part 121 travel.