
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
Elective 4-4: Polar Coordinate Parametric Equations

In the Cartesian coordinate system $xOy$, the curve $C$ is described by the parametric equations $\begin{cases}x=a\cos \phi \\ y=b\sin \phi\end{cases}$ (where $\phi$ is the parameter, $a > b > 0$), and the point $M(2, \sqrt{3})$ on curve $C$ corresponds to the parameter $\phi= \frac{\pi}{3}$. Establish a polar coordinate system with $O$ as the pole and the positive half-axis of $x$ as the polar axis.

$(I)$ Find the general equation of curve $C$ and convert it into a polar coordinate equation;

$(II)$ If $A(\rho_1,\theta_1)$ and $B(\rho_2,\theta_1+ \frac{\pi}{2})$ are two points on curve $C$, find the value of $\frac{1}{\rho_1^2}+ \frac{1}{\rho_2^2}$.

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