
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
Given the polar equation of curve $C$ is $\rho^2= \frac{9}{\cos^2\theta+9\sin^2\theta}$, establish a Cartesian coordinate system with the pole as the origin and the polar axis as the positive half-axis of $x$.

$(1)$ Find the standard equation of curve $C$;

$(2)$ Points $A$ and $B$ are on curve $C$, if $OA \perp OB$, find the value of $\frac{1}{|OA|^2}+ \frac{1}{|OB|^2}$.

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