What is the improvement or update you wish to see?

I'm looking for the correct way to use server actions to update form data (server component form without reloading page)
I can't find anything that looks like this in the server action documentation

Is there any context that might help us understand?

For example:

async function search(data) {
    'use server'
    // ?? do something like redirect or router.push here ???
}

return <form action={search}>...

Does the docs page already exist? Please link to it.

https://nextjs.org/docs/pages/building-your-application/data-fetching/building-forms