I would like to develop a Plutus script that executes each time an epoch ends.
What's the best way to achieve this behaviour?
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As mentioned by Jey earlier, blockchains and smart contracts remain latent until triggered by a user submitted transaction.
One way you could pull this off is that you could use the cardano-cli application to create, sign and submit a transaction which executed the plutus script at the end of every epoch. You can find out more about cardano-cli and plutus scripts here and a simple sample here
Hope this helps!