If you win PUZZLE #71, what's the safest way to sweep the UTXOs?

Hey everyone We all remember how puzzles #66 and #69 got stolen, right? So, if you somehow get your hands on the private keys, what’s the best way to safely sweep those funds?

There is no safe way to broadcast a transaction with largely compromised private key: the moment it hits mempool, multiple bots will start an RBF-fight up to the point where the large majority of the Bitcoins are burned as mining fees. The only way to keep it private until it's confirmed is by using a private mining pool that doesn't broadcast it. MARA Slipstream is said to work, but I haven't tested it. You should post this in Bitcoin puzzle transaction ~32 BTC prize to who solves it.

This has been discussed a a few times in different threads. Just read these threads/posts for the answers; The former contains more information about your options and the latter is an example of the best option you currently have: solve key 66 67 Puzzle how to avoid double spends the tx?/index.php?topic=1306983.msg64379149#msg64379149

That is not the issue here. Those "puzzle" private keys have limited range, #71 specifically is easy enough to be solved in mere seconds once its public key is exposed. The moment you broadcast the transaction to the network, signed offline or not, its public key will be revealed to those thousand "puzzle-stealers" who watch it. And given the info above, it'll just be a game of who's transaction that spends the same input that'll get into a block. It may take a lot of quick unconfirmed txn replacements that could eventually spend the majority of the amount as fee. So even if he somehow wins that, the end result wont be good.

Apparently you have no clue what you're talking about, so why do you post a completely wrong answer when in the post above yours by nc50lc the second link describes a valid solution. To successfully claim puzzle #71 you need to publish and get the redeeming transaction mined non-publicly, e.g. via https://slipstream.mara.com, which to my knowledge has worked fine for puzzles #67 and #68 which both weren't seen in public mempools AND mined by MARA pool. This of course requires trusting the mining service which doesn't seem to be a problem with above mentioned MARA pool service. Puzzle #69 looks very much to have been stolen because the redeeming transaction has been published to public mempools and subsequently was RBF replaced while waiting in public mempools to be mined into a block. There are active bots waiting 24/7/365 that some clueless puzzle solution finder broadcasts the redeeming transaction publicly. In that case there will be a RBF replacement war between bots, desperate solution finder until the last standing replacement transaction gets mined or the prize coins have been converted mostly into transaction fee (the miner benefits!). Stop immediately posting nonsense in this part of the forum or you will very likely be called out.