Getting F group elements form words of standard generators

Let us denote by a, b: [0,1] \to [0,1] the following elements (so called standard generators) of the Thompson F group: a(t)= \begin{cases} 2t & \text{for } 0\leq t\leq \frac{1}{4},\\ t + \frac{1}{4} & \text{for } \frac{1}{4} \leq t \leq \frac{1}{2},\\ \frac{t}{2} + \frac{1}{2} & \text{for } \frac{1}{2} \leq t \leq 1; \end{cases} \qquad b(t)= \begin{cases} t & \text{for } 0\leq t\leq \frac{1}{2},\\ 2t - \frac{1}{2} & \text{for } \frac{1}{2} \leq t \leq \frac{5}{8},\\ t + \frac{1}{8} & \text{for } \frac{5}{8} \leq t \leq \frac{3}{4},\\ \frac{t}{2} + \frac{1}{2} & \text{for } \frac{3}{4} \leq t \leq 1. \end{cases}

Write below a word consisting of a, b or their non-zero integer powers in the LaTeX format.
Example: a^{6} b^{-1} a b^{7}. Spaces are optional, curly braces are necessary. Empty word is allowed.
After successful validation you can plot the element as a piecewise linear bijection g: [0, 1] \to [0,1].
Functions are composed applying first the rightmost one.

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