Back to a function composition question! Thanks <3

For each $x$ in $[0, 1]$, define

$\begin{cases} f(x) = 2x, \qquad\qquad \mathrm{if} \quad 0 \leq x \leq \frac{1}{2};\\ f(x) = 2-2x, \qquad \mathrm{if} \quad \frac{1}{2} < x \leq 1. \end{cases}$

Let $f^{[2]}(x) = f(f(x))$, and $f^{[n + 1]}(x) = f^{[n]}(f(x))$ for each integer $n \geq 2$. Then the number of values of $x$ in $[0, 1]$ for which $f^{[2005]}(x) = \frac {1}{2}$ can be expressed in the form $p^{a},$ where $p$ is a prime and $a$ is a positive integer. Find $p+a.$