
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 a plane Cartesian coordinate system, the curve $C_1$ has the parametric equations $\begin{cases} x = a\cos \phi \\ y = b\sin \phi \end{cases}$ $(a > b > 0,\ \phi$ is the parameter), and in the polar coordinate system with the Origin $O$ as the pole and the positive $x$-axis as the initial ray, curve $C_2$ is a circle whose center lies on the initial ray and passes through the pole. It is known that the point $M(2, \sqrt{3})$ on curve $C_1$ corresponds to the parameter $\phi= \dfrac{\pi}{3}$, and intersects with curve $C_2$ at point $D(\sqrt{2}, \dfrac{\pi}{4})$.
$(1)$ Find the equations of curves $C_1$ and $C_2$.
$(2)$ If $A(\rho_1,\theta)$ and $B(\rho_2,\theta + \dfrac{\pi}{2})$ are two points on the curve $C_1$, find the value of $\dfrac{1}{\rho_1^2} + \dfrac{1}{\rho_2^2}$.

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