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Puzzles = Speed?

Time seems to currently be my main issue (people at my own elo tend to spot the moves quicker), and so I want to know if puzzles are an efficient way to help you speed up your thinking?
Puzzles help analysing situations and help you see patterns which can help you at this. You can also try to play bullet games if you want to train yourself to think faster.
But wouldn't bullet just make me memorize certain moves as an answer to specific sequences whereas puzzles make me understand patterns and actually allows for me to have control of the potential outcome(s) and thus knowing WHAT to respond with and WHY? I'm looking to play faster, but since every opponent play's the game differently, I just feel like bullet would be useless as I'd never come across someone else playing the exact same moves. Besides, I would simultaneously also like to understand what is going on on the board instead of just playing a specific sequence I've memorized and then see the outcome.
@Chezz07 Play classical with increment so you don't have any chance of flagging. Bullet will only get you to pray that something is sound without you actually calculating, so that's not whaat you want. Bullet only breeds bad habits and will not help you improve your chess. I only play bullet for fun, but I would never analyze bullet games or even blitz- too fast.
I've frequently heard grandmasters and experienced chess coaches talk about this. The question comes up in their streams nearly every stream. In the last 2 weeks that includes GM Denes Boros, GM Maurice Ashley, and IM and coach John Bartholomew. They all say exactly the same thing. Spotting good moves depends on pattern recognition and you can't get speed and accuracy without that. To do that they all recommend:

1/. No Bullet
2/. Play longer games, and analyze them afterwards. GM Boros recommends at least Rapid. IM Bartholomew recommends at least 15+10. GM Ashley just said "longer games".
3/. Train tactics.
4/. Go over GM games. Particularly games that have an opening you use, games by GMs who have a similar style to you.
5/. Studying endgames is more important than memorizing scads of opening lines.

My own input: Tactics training should be about accuracy, not speed. If you get one wrong it important to understand why your solution is wrong before you move on to the next problem. I recommend chesstempo.com for tactical training. It's much tougher then lichess' puzzles (my tactics rating there is about 600 points lower than here). But it gives a much wider variety of tactical themes.

Seems the consensus is speed with accuracy comes faster with practicing the skills, not practicing rushing. Then building speed once the skills are there. Age also makes a difference here. I'm much slower than I used to be back when I was in my 20s or early 30s.

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