A trader creates a long butterfly spread from options with strike prices X$,$ Y$,$ and Z$,$ where X $<$ Y $<$ Z$,$ and Y is exactly midway between X and Z$.$ A total of $400$ options are traded. The difference between X and Y is $$11.$ The difference in the prices of the options with strike prices of Z and Y is $$5\mathrm{.}07.$ The difference in the prices of the options with strike prices of Y and X is $$6\mathrm{.}96.$ What is the maximum net gain $($after the cost of the options is taken into account$)?$ Please give your answer to the nearest dollar.

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