Turkish Airlines' Staff Provided Incorrect Information Leading Me to Missed Flight

Hello, I arrived at the airport at 8:00 AM for my flight scheduled at 8:30 AM on November 7, 2023. I obtained the necessary tickets and was at the gate around 8:10-8:15 AM. However, the flight attendant did not allow me to board, even though my brother had boarded just 2-3 minutes before me, resulting in us narrowly missing the flight. Since my ID was with my brother, he had to disembark as well. Consequently, my mother, brother, and I missed the flight due to a mere 2-3 minute delay. We immediately approached the staff to inform them that we had just missed the flight and asked if there was anything that could be done. The staff member informed us that nothing could be done and that we would need to purchase new tickets. Later, when I spoke with my sister, I learned that in case of a missed flight, we could purchase tickets at a much lower price by paying a penalty fee. Armed with this information, I returned to the staff member, who then directed us to a different ticket sales point. We spoke with the staff there around 8:38 AM. This staff member informed us that if we had arrived by 8:30 AM at the latest, they could have processed the penalty ticket, otherwise, we would have to buy new tickets. If the first ticket sales staff had not misled us, we would not have incurred such a financial loss (the new tickets cost a total of ₺2700). Due to the financial loss caused by the staff's incorrect guidance, I request a refund for the new tickets I had to purchase.