Flight delayed for 24 hours. What are my rights?

Lufthansa changed my itinerary and I’m flying now from Cali, Colombia to Berlin, Germany 24 hours later than what I booked. I reached out to their customer care and they didn’t offer alternative flights (they say there are none for the day of my original flight or before), they won’t help me with finding accommodation for the extra day, and they don’t guarantee to reimburse any of my costs.

What can I do and what can I demand from them? I will miss a day of work because of the delay as well.