
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
In the rectangular coordinate system $(xOy)$, establish a polar coordinate system with the coordinate origin $O$ as the pole and the positive semi-axis of $x$ as the polar axis. The polar coordinate equation of curve $({C}_{1})$ is $ρ(ρ-4\cos θ)={r}^{2}-4$, and the parametric equation of curve $({C}_{2})$ is $\begin{cases}x=4+\sqrt{3}r\cos θ\ y=\sqrt{3}r\sin θ\end{cases} (θ$ as the parameter$).\;$

(1) Find the rectangular coordinate equation of curve $({C}_{1})$ and the polar coordinate equation of curve $({C}_{2})$;

(2) When $r$ varies, the trajectory of the intersection point $M$ of $({C}_{1}{C}_{2})$ is curve $({C}_{3})$. If a straight line $l$ passing through the origin $O$ with an inclination angle of $\frac{π}{3}$ intersects curve $({C}_{3})$ at points $A$ and $B$, find the value of $|OA|-|OB|$.

Present the answer in LaTex format: \boxed{Your answer}
user