Hello,
If the issue is still on the same user that set status to Done, you can use the "assign to current user" function of post-functions on the reopen transition of the workflow.
If the issue is not on the same user that set status to Done, you can check out this plugin that offers auto-reassignment.
https://marketplace.atlassian.com/plugins/ru.andreymarkelov.atlas.plugins.utils/server/overview
It's possible, just add a JIRA Workflow post-function that can you do what you want but your options is limited since you are using the Cloud version. Check what add-ons are available there for post-functions.
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