If (hypothetically) you found the private key for puzzle 66, and simply tried to spend puzzle 66's BTC balance to another wallet that you control, you better hope that that transaction gets mined into the next block
within a few seconds, or it will be stolen with 100% certainty. This is also the case for the next few puzzles, 67,68,69...
The only way to do that is by using the MARA Pool Slipstream. The transaction will be mined before the public key enters the mempool.
It won't actually be seen at all through the mempool.

Interesting, as I'm not in the key hunting business, I didn't know about
https://slipstream.mara.com/ yet. Thanks for pointing me to it. They seem to charge about double the current fee (as of writing this mempool.space indicated 5 sat/vB and Slipstream asks for a minimum fee of 10 sat/vB; I expected worse).
I've read their ToS briefly and it seems at first glance that indeed they claim your submitted transaction remains "confidential" until mined into a block by their own pool.
One can only hope that slipstream.mara.com are really honest and do not exploit the internal disclosure of the public key themselves. I heard that mara.com does attract some critisism, at minimum as far as I recognized for censoring certain transactions which got tagged as "not appropriate". Can't say anything how reliable they are when at minimum 6.6
BTC are at stake.