A step function f(x) is defined on the interval [0, 1)

From [0, 2/3), the step is 4/5 high.

From [2/3, 8/9), the step is 24/25 high.

From [8/9, 26/27), the step is 124/125 high.

From [1 - 3^-k, 1 - 3^-(k+1)), the step is 1 - 5^-(k+1) high.

What is the exact area under the step function?