Parameterize a union of circles

Homework Statement

Let $C=\lbrace(x,y) \in R^2: x^2+y^2=1 \rbrace \cup \lbrace (x,y) \in R^2: (x-1)^2+y^2=1 \rbrace$. Give a parameterization of the curve C.

The Attempt at a Solution

I'm not sure how valid it is but I tried to use a 'piecewise parameterisation', defining it to be $r(t)=(cos(t+\frac{\pi}{6}), sin(t+\frac{\pi}{6})$ for all $t \in [0, 2\pi)$ and $r(t)=(cos(t+\frac{2\pi}{3})+1, sin(t+\frac{2\pi}{3})$ for all $t \in [2\pi, 4\pi]$.

So it traces out the first circle out at the top intersection and then the second. But I'm not sure how to make this smooth at the intersection, and one-to-one, as I'm quite sure it has to be. This is the best I could come up with.

SammyS
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Homework Statement

Let $C=\lbrace(x,y) \in R^2: x^2+y^2=1 \rbrace \cup \lbrace (x,y) \in R^2: (x-1)^2+y^2=1 \rbrace$. Give a parameterization of the curve C.

The Attempt at a Solution

I'm not sure how valid it is but I tried to use a 'piecewise parameterisation', defining it to be $r(t)=(cos(t+\frac{\pi}{6}), sin(t+\frac{\pi}{6})$ for all $t \in [0, 2\pi)$ and $r(t)=(cos(t+\frac{2\pi}{3})+1, sin(t+\frac{2\pi}{3})$ for all $t \in [2\pi, 4\pi]$.

So it traces out the first circle out at the top intersection and then the second. But I'm not sure how to make this smooth at the intersection, and one-to-one, as I'm quite sure it has to be. This is the best I could come up with.
What are the coordinates of the points of intersection ?

haruspex
@SammyS: The circles intersect at $x=\frac{1}{2}$