Do miners keep a "balance" of how much each public key has?

Do miners keep a "balance" of how much each public key has?

I am trying to understand how double spending can be prevented if mining nodes do not keep a current "balance" of BTC associated with an account?

If they do keep the current balance isn't this a problem as the number of potential accounts is very, very large? That is, if each account balance takes up space in memory, and more accounts are created then the only way nodes can keep up is when the rate of growth of accounts is less than the rate of growth of available memory per node?

Am I missing something?

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