This is the flip side of #197, as part of this Zulip feature: https://zulip.com/help/status-and-availability
API docs: https://zulip.com/api/update-status

#197 is a prerequisite, so that we have the current value of the self-user's own status.

Design

In the UI, this will be its own screen, reached from the "My profile" screen. For details, see Figma, starting here:
https://www.figma.com/design/1JTNtYo9memgW7vV6d0ygq/Zulip-Mobile?node-id=5000-52611&m=dev
and in the screens to the right of that.