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The problem might be that the solenoid is sticking and does not open the oil passage to activate the reverse gear. The transmission oil has a tendency to form a sticky substance that cause this to happen. To remedy this would entail removing the transmission oil plan along with the solenoid bank and clean them and or replace if they are faulty. At times they burn/overheat and become faulty. You might very well also have fault codes stored in the ECM about this problem.
Will also be a good idea to replace the oil filter on the inside of the transmission as they do become clogged.
This should answer your question but should you need more information and or assistance please do not hesitate
I really do not understand why the people in your country ask so much money for the job as they can do this in and around one and a half hours time.