A gambling game involves a fair die with six faces. Each face is marked with a distinct number of dots from one and six. You pay a fee to play every round of the game. The fee for the first round is ₹0.01. The fee increases by ₹0.01 after each round is played. Thus, the fee to play the second round is ₹0.02, the fee to play the third round is ₹0.03, and so on. Once you have paid the fee, it won't be returned to you.
In each round, after you pay the fee, the die is rolled and you win ₹1.00 times the number of dots that appear on the face facing up. You may quit the game at any moment. How long should you play this game if you want to maximize your profit? What is the profit you expect to make?